When you say 'Dravidians in south India', you've probably assumed a huge misconception right there. To say India has a dark-skinned populace exlusively in the south and fair skinned ones in the north is a fallacy. Over millenia, large numbers of the northern Aryans have migrated to the south assimilated the local culture, and similarly too large populations of Dravidians have either remained firmly in the north or have migrated to, there. For example, in India's most southern state, Kerala, although the general population is dark skinned, there are large swathes of the state with fair skinned, Dravidic language speaking people, with skin lighter than their northern Indian contemporaries. This is evidence for the Aryan migration into all over India, and not just the south.
Another misconception is that of Dravidic languages being classified as such, distinct from indo-european ones. But the fact is ,each of these southern, Dravidian languages have about 50+ % Sanskrit origin, which is an Aryan (Indo-European) tongue.
Here's one famous south Indian singer...
... observe the general eye and skin colour of this 'dravidian'.