South Indians are different race (Dravidians), speak an entirely different language family (Dravidian), have different genetics and appearance. Even the Hindu religion they practice is different from North India. So, how and when did South India become part of India?
The Aryan Invasion Theory is by no means proved beyond doubt. There is a lot of ongoing dispute on it to this day.
They don't have different genetics. There is very little mtDNA variation between north and south Indians.
The Hindu religion is different across the country and has always been.
Sorry but your question is not exactly accurate