The origin of the Tamil dynasties is quiet old and Tamil /Tamils are referred to in the Ramayana, Mahabharatha, Vishnu Purana, Sanskrit literature, for example in Kalidasa’s Raghuvamsa.

The Pandyas trace their ancestry to Lord Shiva whom they considered to be their Family Elder.

Cholas trace themselves to Suryavamsa, Solar Dynasty to which Rama belonged and some records indicate they trace themselves to Sage Kasyapa.

Chera dynasty seems to lie with Ila,the term Elam seems to be from this word(daughter of Vaiwastha Manu, the ancestor of Lord Rama, the First Human, who migrated to Ayodhya due to a Tsunami in South India.

Manus son Ikshvaku founded the Suryavamsa , the Solar dynasty from Ayodhya.

The Cheras seem to belong the female lineage of Manu.

Now, there is another dynasty which was powerful.

IT was the Pallava Dynasty which ruled from Kanchipuram,Tamil Nadu.

The Mahabalipuram shore temples are the handiwork of Mahendra and his son Narasimhavarma Pallava.

Simha vishnu was the father of Mahendra Varma Pallava.

The inscription in Amaravathi Andhra refers that Simhavishnu was from the lineage of Aswathama.

‘But interestingly, the Pallava kings of Kanchi, had this legend about their genealogy, described in an inscription found near Amaravathi, Andhra, from the reign of Simhavarma Pallava,

According to which, a child was born to Aswattama and an celestial dame (apsara?) in the woods and was abandoned in a bed of creepers. And that child eventually started the dynasty.

The words for “creepers” is Pallavam, Pouthram in Sanskrit and in Tamizh, “thulir, thondai”. Hence the Pallavas aka Botharasas aka Thondaimaans.‘

Reference. https://www.quora.com/Did-Ashwathama-from-the-Mahabharatha-marry

There is another reference to the Pallava Dynasty in the old Tamil Epic Manimekalai , which is one of the Five Epics of Tamil, which states that the Thondamans were the descendants of Aswathama.

Cholas are also traced this way.

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The earliest documentation on the Pallavas is the three copper-plate grants, now referred to as the Mayidavolu, Hirahadagalli and the British Museum plates (Durga Prasad, 1988) belonging to Skandavarman I and written in Prakrit. Skandavarman appears to have been the first great ruler of the early Pallavas, though there are references to other early Pallavas who were probably predecessors of Skandavarman.Skandavarman extended his dominions from the Krishna in the north to the Pennar in the south and to the Bellary district in the West. He performed the Aswamedhaand other Vedic sacrifices and bore the title of “Supreme King of Kings devoted to dharma”

In the reign of Simhavarman IV, who ascended the throne in 436, the territories lost to the Vishnukundins in the north up to the mouth of the Krishna were recovered.The early Pallava history from this period onwards is furnished by a dozen or so copper-plate grants in Sanskrit. They are all dated in the regnal years of the kings.

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attributes the origin of the first Pallava King from a liaison between the daughter of a Naga king of Manipallava named Pilli Valai (Pilivalai) with a Chola king, Killivalavan, out of which union was born a prince, who was lost in ship wreck and found with a twig (pallava) of Cephalandra Indica (Tondai) around his ankle and hence named Tondai-man. Another version states that “Pallava” was born from the union of the Brahmin Ashvatthama with a Naga Princess also supposedly supported in the sixth verse of the Bahur plates which states “From Ashvatthama was born the king named Pallava”. The Pallavas themselves claimed to descend from Brahma and Ashwathama.

Though Manimekalai posits Ilam Tiriyan as a Chola, not a Pallava, the Velurpalaiyam plates dated to 852, do not mention the Cholas. Instead, they credit the Naga liaison episode, and creation of the Pallava line, to a different Pallava king named Virakurcha, while preserving its legitimising significance:

…from him (Aśvatthāman) in order (came) Pallava, the lord of the whole earth, whose fame was bewildering. Thence, came into existence the race of Pallavas… [including the son of Chūtapallava] Vīrakūrcha, of celebrated name, who simultaneously with (the hand of) the daughter of the chief of serpents grasped also the complete insignia of royalty and became famous.

Historically, early relations between Nagas and Pallavas became well-established before the myth of Pallava’s birth to Ashvatthama took root