What keeps pushing India northward into Asia, tens of millions of years after the two collided?

Answer: Sinking slabs of plate and flowing rock deep below keep pulling it along

Sinking slabs of plate and flowing rock deep below keep pulling it along ✓ — India is pulled and dragged from below: cold, dense slabs of plate sinking into the mantle tug on it, and slowly flowing mantle rock drags on its underside. Both forces are still acting today, which is why the plate keeps moving 4 to 5 centimetres a year; geologists still debate which of the two does more of the work.

Leftover momentum from India's fast drift before the collision — A tectonic plate has almost no momentum to coast on. India moves only a few centimetres a year, so the forces on it balance at every moment, and it would stop almost at once if nothing kept pulling or dragging it. Roughly 50 million years of continued motion therefore needs a force that is still acting today.

The weight of the Himalayas dragging India forward beneath them — The weight of the Himalayas and Tibet pushes back against India instead of pulling it in. High ground presses outward on its surroundings, and that resistance helped slow India from about 15 centimetres a year before the collision to about 4 or 5 afterwards. Whatever drives India has to overcome it.