No, there are no genetic similarities found between these two populations. Here I am going to compare the Dravidian movement with Anglo-Saxons’ rebellion against the ruling Anglo-Normans to herald an egalitarian English society during middle ages.
Divisions in the society:
The following is an account given by few historians but I'm not sure of its objectiveness. After capturing
The Dravidian movement maintained that they Dravidians were casteless society until the arrival of Brahmins. The movement was solely against Brahmins (who also had become Dravidians linguistically and culturally). Isn’t it striking that Dravidians could be equated to Anglo-Saxons and Brahmins to Anglo-Normans? Anglo-Saxons might have been classless but not so when it comes to Dravidians.
Dravidian classes and castes:
Manu(author of Manusmriti) had declared that all Dravidians were Shudras. Understandably, Brahmins divided the society into pure-shudras and lowest shudras(not really ‘untouchables’) with minuscule population passing themselves as Kshatriyas and Vaishyas. However, how did they get the people to classify as ‘Asprishyas’? It shows though Dravidian society was casteless, it was obviously class ridden. A sizeable number of Dalits have caste names like Adi Dravida, Adi Karnataka and Adi Andhra, implying that they might be the first inhabitants of
2. Right hand and left hand castes:
It is not clear whether these divisions existed in the Dravidian society prior to the entry of Brahmins or after them. This division had no affiliation to the caste divisions. People belonging to the same caste might be part of either of these two. However, I don’t think this division has anything to do with ‘right hand path’(Dakshinachara) and ‘left hand path’(Vamachara) practices found in Vaidik and Buddhist religions. The religious practices of right hand caste and left hand caste do not show any differences. I belong to a left hand caste and I have not seen or heard from elders, the typical Vamachara practices. But the main point here is these two groups used to fight till 19th century. And it’s still a big mystery why ever they fought? Also, as far as I know, only Tamil and Malayalee societies were divided along those lines. Anyway, it’s still a unique feature of Dravidian society though obsolete nowadays.
Not so egalitarian movement:
If you consider these inherent class structures in the Dravidian society even without Brahmins, this movement was never going to be anything remotely revolutionary. It only gave hegemony back to shudra Dravidians who anyway held sway before the arrival of Brahmins and that continued during Sanskritization albeit without any respect to go with their position.
Why only in Tamilnadu?
One of the founding fathers of this movement was a malayalee, T.M. Nair. However, this movement was successful only in Tamilnadu and never gained any momentum in other Dravidian states. Possible explanations could be,
1. There was no non-brahmin intelligentsia in these states. Tamil Brahmin and non-brahmin Vaishnavites and Shaivites did fight for religious supremacy through the centuries.
2. The Buddhist and Jain Dravidian intelligentsia in Andhra Pradesh, Karnataka and Kerala either got absorbed in the Brahmanical religion or became extinct as it lost political patronization(which generally favoured Brahmins).
3. The Shaivites of Karnataka had their own independent religious movement (Veerashaiva). So they were not really bothered about Brahmins to identify themselves with Dravidian movement.
4. Even though Dravidians in other states did produce intellectuals, the shudra Dravidians had gained prominence in their respective states within few decades of independence. And there was no strong reason to fight against Brahmins.
Not surprisingly, Dalits in Tamilnadu were hardly emancipated with this movement. In some cases, Dalit parties and Dravidian parties ended up in opposite camps. And it must be noted that atrocities against Dalits were more widespread in Tamilnadu than in any other Dravidian states.
In a nutshell, Dravidian movement was as non-dalit as it was anti-brahmin. The Dravidian society even without Brahmins was never classless. So the movement was never a movement of enlightened.
1. Adi, first in Sanskrit
2. Nairs in Kerala were the dominant feudal caste.
1. Caste and Race in India by G.S. Ghurye