There’s an old saw that teachers would sometimes use, that if there were a tubelike tunnel straight through the Earth, an object which was dropped in would oscillate down one side, up the other, until it finally stopped dead and hovered at the center of the Earth. It’s not true. :^)
So what would happen
What would a actually happen I suppose would depend on the mass and shape of whatever object is dropped. Suppose it were a perfect sphere weighing ten pounds, cropped into a hole twice the diameter of the sphere?
My hunch is that it would stop at the center (taking only gravity into account and not the forces caused by heat).