There are many historical events of Indo-European speakers invading India from the north, like Darius the Great, Alexander the Great, the Mughals. AIT refers to a theory from an earlier time period but rejecting it does not rule out subsequent Indo-European conquests of the region which surely contributed to the spread of Indo-European culture and language southward.
That doesn't explain Sri Lanka which is past South India...
Edit: The actual answer is North/Eastern Indians speaking an IE variant immigrated to Sri Lanka in the 5th/6th centuries and became the dominant language there since (Sinhala)