Assuming enough of the body remains a solid near the core (highly unlikely):
Wind resistance would be the only thing slowing the body as it approaches the center. An object does not slow down because gravity decreases, it only accelerates more slowly.
At some point the body would reach terminal velocity; the point where wind resistance counters gravity's acceleration. As gravity decreases that point will get slower and slower, however it would not be zero at the center. How fast the body would still be going I don't have the math to make an accurate (or even semi-accurate) guess. It would keep going until the the now rising gravity overcame the remaining velocity. At that point you'd have the same situation in reverse, with less energy. And you'd bounce back and forth a bit like a rubber band, going a bit slower with each cycle until you ended up at the center anyway.
If you eliminated the wind resistance and the heat, you'd could bounce back and forth (end to end) forever, or there abouts. There is always some energy loss in any system.