Imagine a tiny black hole arriving on earth. It grabs some mass. The black hole is moving, but the mass it ate is not. So rather than free falling, it looses momentum. As it falls through the earth it keeps doing that.
So it will not rebound back up to the top at the same height it fell. It will be lower. The second time through the same thing will happen.
Each time it oscillates it will get lower and lower.
Next. Say the black hole is a bit bigger. And it eats some mass on one side of the earth - now there is an imbalance. There is more mass on the other side. So the forces will cause it to slow down more than it otherwise would.
When it finally settles down enough to make a cave, it will "swing" as much as it can. But each swing can not be any higher than the one before. Over time, with random atoms falling into the cave it, it will come to rest at the exact center.
There is no force on the black hole, so if it moved at all it would crash into the wall. Only when it actually ate some of the wall would it turn around. So each swing would eat more and more.
> You are missing three points: one is that if there is no cave, and it's oscillating through the earth, it will be pulled to the center.
Agreed.
> Two is that it looses momentum each time through the earth.
I'm not so sure about that. Mass is added to the black hole, and perhaps this causes the total momentum to increase.
> And three is that once it's inside it's hollow cave it can no longer reach the wall.
If it is stopped inside the cave. But there is no force to stop it.