Below is in full, the chapter on Asmakas from Ancient Indian Tribes by Bimla Churn Law(what the hell is his actual spelling?). This piece of writing gives us the maximum information over Asmakas.

The Asmakas or Assakas formed one of the Ksatriya tribes of ancient India. They are not mentioned in the Vedic literature, but we find them referred to in the Epics and the Puranas. In the enumeration of the countries in Bharatavarsa, the land of the Asmakas is mentioned along with those of the most prominent Ksatriya peoples of ancient India, viz., the Kurus, Surasenas, etc. (Bhlsmaparva, Ch. 9, p. 822 ). In the different recensions of the Mahabharata, the name is spelt in different ways, viz., Asvaka or Asmaka. In Buddhist literature the name is Assaka, which, as Prof. Rhys Davids points out, may be the vernacular equivalent of either Asmaka or Asvaka. The Professor observes, “The name of the tribe is ambiguous. Sanskrit authors speak both of Asmaka and Asvaka. Each of these would be Assaka, both in the local vernacular and in Pali. Either there were two distinct tribes so called, or the Sanskrit form Asvaka is a wrong reading or a blunder in the Sanskritisation of Assaka.”. The Greek writers mention a people called the Assakenoi in eastern Afghanistan and the Khonar Valley, with their chief town at Massaga or Masakavati. It is difficult to say whether they were identical with our Asmakas.
In the Great Epic there is some confusion between the Asmakas and the Asvakas; some of the passages appear to contradict one another. In the Jayadrathavadhaparvadhyaya, the Asmakas are found ranged on the Pandava side (VII. 85,3049); on the other hand, an Asmakadayada, or a son of the Asmaka monarch, is said to have been killed in battle by Abhimanyu (VII. 37, 1605); and the same person is also referred to as Asmakasya suta in the verse immediately following (VII. 37,1606). An Asmakesvara is also spoken of here (VII. 1608). In a list of the tribes conquered by Kama, the Asmakas are mentioned along with the Vatsas, Kalingas, Rsikas, etc. (VIII. 8,237). In the Adiparva, a Rajarsi Asmaka, the son of Vasistha and Madayanti the wife of Kalmasapada, is mentioned, and the story of his birth, which we shall speak of in great detail hereafter is referred to. (I. 122,4737). The same king who is called a Vasistha is said to have founded Paudanya (I. 177,6791). Panini mentions Asmaka in one of his sutras (IV. 1,173).
The Anguttara Nikaya, like the Puranas, tells us that Assaka was one of the sixteen mahajanapadas of Jambudipa. It had abundance of food and gems. It was wealthy and prosperous. From the Mahagovinda Suttanta we learn that Potana was the city of the Assakas. It was undoubtedly the capital city as King Brahmadatta reigned there. Asanga in his Sutralankara mentions an Asmaka country in the basin of the Indus. From this reference it would appear that there was an Assaka country in northern India, but in Buddhist literature we also read of a southern Asmaka country. Thus one of the oldest works of the Pali Buddhist literature, the Sutta-Nipata (verses 976-7) speaks of a Brahman guru called Bavari, who having left the Kosala country, settled near a village on the Godavari in the Assaka territory in the Daksinapatha (D. R. Bhandarkar, Carmichuel Lectures, 1918, p. 4). Again, in the Sutta Nipata (verse 977) the Assaka or Asmaka country has been associated with Mulaka with its capital Patitthana, and mentioned as situated immediately to the south of the latter but along the river Godavari, as Dr. Bhandarkar points out (Ibid. p. 53, n. 5). Evidently the Asmakas, or at least an offshoot of the tribe, had settled in the south on the banks of the Godavari.
Dr. Rhys Davids points out that the country is mentioned with Avanti in the same way as Anga is with Magadha and its position on this list (the list of the sixteen Mahajanapadas), between Surasena and Avanti, makes it probable that when the list was drawn up, its position was immediately north-west of Avanti. In that case the settlement on the Godavari was a later colony and this is confirmed by the fact that there is no mention of Potana (or Potali) there (Buddhist India, pp. 27-28).
We had already referred to the story of the origin of Asmaka, the founder of the tribe, as mentioned in the Mahabharata. But in the Great Epic there is a bare reference to the story which is fully narrated in the Brhannaradiya Purana. Once Sudasa, who is often identified with the Great Rgvedic hero who won the battle of the ten kings, the great grandson of Rtuparna, the seventh in descent from Bhaglratha went to the forest for hunting. He killed a tiger. The dying tiger took the shape of a terrible monster and thought of wreaking vengeance on the king. An occasion soon presented itself. King Sudasa performed a sacrifice. When Vasistha, the king’s priest, departed after performing the sacrifice, that monster assumed the form of Vasistha and said to the king, “Feed me with meat to-day. Prepare it, l am coming back”, and then went away. The monster once more changed his appearance and appeared before King Sudasa in the guise of a cook. He cooked human flesh when ordered by the king to prepare a dish of meat for the great Rsi. The king waited for Vasistha with the cooked meat on a golden plate. When the genuine Rsi Vasistha came, Sudasa offered him that meat. Vasistha took him to be a very wicked king who could go so far as to offer him meat. Then he meditated and learnt that it was human flesh dressed up for him. He cursed the king, saying, “Knowing it to be human flesh you have offered it to me, so you will be a monster greedy of human flesh”. King Sudasa said that he had done so by his order. Vasistha sat in meditation, learnt everything and said, “You will have to remain a monster only for twelve years and not for ever”. The king was about to curse Vasistha but Madayanti, his queen, entreated him to forbear and appeased his wrath. The king washed his feet with the curse-water. His legs turned black. Thenceforward he was famous as Kalmasapada. Every third night the king took the shape of a raksasa and strolling about in the forest used to kill human beings. One night in spite of the requests of a Brahmani, he ate up her husband. The Brahmanl cursed him, “You will die at the time of union with your wife”. After the expiry of twelve years the king was freed from the curse of Vasistha. The king recollected the curse of the Brahmani and refrained from approaching the queen Madayanti. At his request Vasistha caused the conception of the queen. Seven years elapsed but delivery did not take place. The Queen Madayanti struck the womb with an “asma”, or a piece of stone, and a son was born who was named Asmaka. Asmaka’s son was Mulaka. Having been saved by naked women who surrounded him, he was named Narikavaca. His great grandson is said to have been Dilipa, the forefather of the famous hero of the Ramayana. Thus a connection is established between the Iksvakus and the Asmakas (Brhnnaradiyapurana, Ch. 9).
In the Bhavisyapurana also Asmaka is mentioned as the son of Sudasa. It is probable, as we have suggested before, that the Assakas were an offshoot of one of the great Ksatriya families of the early times.
The Matsya Purana (Ch. 272) gives us a list of twenty five Asmaka kings, contemporaries of the Sisunakas, who reigned in Magadha before the Nandas. Apparently, about this time the Asmakas had risen into prominence and taken their place beside the royal dynasties of northern India.
One of the Jatakas relates the following story about a king Assaka. In Potali, the capital of Assaka there reigned a king Assaka. He had a queen of unique beauty. At her death, the king was overwhelmed with grief. At this time, the Bodhisatta dwelt at the foot of the Himalayas. With his heavenly vision he saw the king lamenting, and moved to pity; he came to a park where he met a young Brahmin who told him that the king was lamenting the loss of his queen. The Bodhisatta said that he could show the king his queen and even make her speak to him. The young Brahmin informed the king who hastened to the spot. The Bodhisatta showed him his queen who after death was leading the life of a tiny dung-worm. Upon the king making himself known to his whilom beloved queen, the dung-worm told him in human voice that she no longer loved the king; for dearer to her was the worm. The king was astonished. The Bodhisatta instructed him and left the place for the Himalayas (Cowell, Jataka, Vol. II, pp. 108-110).
Another story of the Assaka country and its connection with Kalinga is narrated in the Jatakas. Assaka was the king of Potali in the Assaka country. At this time Kalinga was reigning in the city of Dantapura in the Kalinga kingdom. Kalinga had four daughters of surpassing beauty, whom he ordered to sit in a covered carriage to be driven to every village, town and royal city with an armed escort. Kalinga declared that if any king would be desirous of taking them into his harem, he would put up a fight with him. Passing through various countries, they reached Potali in the Assaka country. The gates were closed against them, but were opened by order of Nandisena, the able minister of the king of Assaka. The four princesses were brought to the king who was asked by his minister to make them his chief queens. Accordingly, these fair princesses were raised to the dignity of queen-consorts and a message was sent to Kalinga. King Kalinga, on receipt of the message set out with a great army and halted within the limits of his own territory and Assaka also kept within his. A great battle was fought. Through the diplomacy of Nandisena, Assaka defeated Kalinga who then fled to his own city. Assaka demanded from Kalinga a portion of the dowry received by his daughters who were royal maidens. Kalinga sent a befitting portion of it for his daughters to Assaka. Thenceforth the two kings lived amicably (Cowell, Jataka, III, pp. 2-5). This story shows that the Assakas and the Kalingas were neighbours and that their countries bordered on each other. Evidently, it is the southern Assaka country on the Godavari that is here referred to.
The Vimanavatthu commentary tells us a story of an Assaka king who was ordained by Mahakaccayana. In the kingdom of Assaka, there reigned a king named Assaka whose capital was at Potananagara. He promised to grant a boon to his younger wife. When his son named Sujata by his first wife, was sixteen years of age, his younger wife reminded him of his promise and prayed that Sujata should be banished and sent to a Forest and her son should succeed him to the throne. The king was vacillating, but at last Sujata was sent to a forest where he met Mahakaccayana in a hermitage. Being instructed by Mahakaccayana in Dhamma, he became a bhikkhu afterwards (Vimanavatthu Commentary, pp. 259 full).

There are a few takeaways. Though we are seeing such a long chapter, the information conveyed regarding Asmaka is literally zero. No clarity as to where the Asmakas are – the author refers to three different ones – one in Afghanistan, one ruling from Potali(modern Bodhan in Telangana) and one north of Paithan. This increases the confusion as to which is the real Mahajanapada. Next, there are two different strains of history in the article. Hindu and Buddhist. The Hindu ones give a garbled version of some kingdom called Asmaka and a person called Asmaka while Buddhist ones give details about a kingdom called Asmaka, based in Potana, may be, a small one, but one which is important in Buddhist parlance. The Hindu versions show a clear attempt to legitimize a small and obscure kingdom as it rose in power by linking it to the Ikshwakus. Did that Asmaka exist, or was he a son of Sudasa, who knows? And if he is, why did the king go that extra length to beget one when someone else ascended the throne after him? Another allied kingdom is legitimized by saying Asmaka’s son is Mulaka. And since they ruled from Paithan, does it imply that the Satavahanas were the servants of Mulakas as against Asmakas and replaced them? Out of respect for their predecessors, the original Andhras, did they take the name Andhrabhrityus? An important thing to note is that, Chapter 273 of Matsya Purana uses the word Sisukondhrah Sajateeyah – Sisuka of the race of Andhras/Sisuka of the kingdom of Andhras – meaning he is identified with Andhras, some race ruled by someone before him. Another interpretation can be that they are the vassals of Asmakas or some other kingdom attributed to Andhras who toppled the Mulakas; Andhrabhrityu being a derogatory/honorific title attributed to their origins.
Now coming to the Matsya Purana itself. Many people, may be, taking this chapter as a source declared Chapter 272 of Matsya Purana gives a list of 25 Asmaka kings, but, no names are given.

कलिङ्गाश्चैव द्वात्रिंशदश्मकाः पञ्चविंशतिः।
कुरुवश्चापि षट्विंशदष्टाविंशस्तु मैथिलाः ।। २७२.१५

Then, there will be 32 Kalinga kings, 25 Asmaka, 26 Kurava and 28 Maithila

Now, who are historic Asmakas? Dandin’s Dasakumaracharita has it’s last chapter as Visrutacharita, over the adventures of one of those Kumaras by name Visruta. That chapter states thus –
When the king of Vidarbha passed away, his young son ascended the throne. He was an expert in everything but was not interested in the science of politics. How much he was told, he never listened to learn the subject but spent his time in pleasures. Seeing him, all the subjects started leading an idle life. Seeing the situation, the feudatory king of Asmaka sent his son to the Vidarbha court to make the kingdom more morally corrupt. When the country was thoroughly disorganized, the king of Asmaka egged the ruler of Banavasi to invade and occupy the kingdom of Vidarbha. The stage was set for a battle on the banks of Varada with the Vidarbha armies consisting of the royal army and that of the feudatories of Asmaka, Kuntala, Murala, Rishika, Nasikya and Konkana. The Asmaka king conspired to cause disaffection in the feudatories who then attacked their suzerain from the rear. The king was killed in battle. He then caused dissensions among the feudatories. They fought with one another over spoils of war and destroyed one another. After buying off the king of Banavasi, the ruler of Asmaka himself occupied Vidarbha and ruled from there.
This tells that the Asmakas of history are feudatories of the Vakatakas who helped in bringing down the Vakataka rule for good. Who were they, where did they come from, are they the same as those historic Asmakas, there is no way to know. Another reference comes from Cave XVII Inscription from Ajanta. It gives a list of nine rulers, Dhritarashtra, Harisamba, Saurisamba, Upendragupta, Agaja or Kacha I, Bhikshudasa, Niladasa, Kacha II, Krishnadasa and Ravisamba. The inscription also states Krishnadasa’s two sons of which Ravisamba is the younger one conquered Asmaka and due to the premature death of Ravisamba, his brother, the king turned an ascetic. Note that they conquered. Now, who are the original rulers of Asmaka? How are they related to the king of Asmaka who brought down the Vakataka Empire – ancestors or conquerors or plain raiders? The naming of Harisena, the king whose death brought forth the chain of events in which his son, the ruler of Vidarbha is killed because of a scheming Asmaka king simply points to the fact that this is either one of the feudatories of Vidarbha like Asmaka or is some kingdom which attacked Asmaka but is brought to book by Harisena. Note that Ravisamba died an unnatural death at a young age (killed in battle? Ambushed?)