Where is the scientific evidence that the people of IVC picked up and shifted en masse to the south?
In the absence of conclusive genetic evidence to the contrary, the logical approach is to treat any group as static, albeit with immigrant infusions over time. We also have absolutely no issue with having incorporated various waves of migration. That is the norm in human civilizations.
You are arguing backwards from a predetermined conclusion. Political boundaries are always fluid and have changed numerous times over the millenia. Modern political boundaries do not constrain history. Whatever justifications were or were not made for this particular setup is irrelevant. We are not saying we are inheritors of IVC because we are Muslim -- that claim is just as wrong as the claim that we can't be. We are making that claim because the land is where it is; the people are who we are; and the history of the land and the people remains inviolate regardless of shifting political boundaries.
You are right, it just so happens that the current State of Pakistan happens to encompass most of the ancient IVC and the land of Panini, etc. But so what? We are the people who have lived here for millenia. That history is ours and would be ours regardless of political boundaries. Just as the history of ancient Tamils belongs to the Tamil people in perpetuity, whether they are part of India or a separate country.
We've been through this. You are grasping at straws to argue backwards from your conclusion.
The Greeks, Swedes, Egyptians and Brits have changed faiths and are perfectly legitimate inheritors of the ancients. Just because you want to claim exclusivity is irrelevant to us. We don't need to be constrained by your self-serving rules which don't apply across the world anyway.