There is a special type of node called slasher that can submit proof of your voting on a different chain (your vote is signed by you, so it could be verify that you sign something not on the current chain), then you lose your stake on the chain. Supposedly both chain should have their own slasher nodes, so eventually you will lose your stake on both chains.

Don't cite me on the details, but that is the idea

But after the fork I move my coins to a new address. How do they know then?

Assume I could have obfuscated the move through the many means availible.

Your stake is locked for a non trivial amount of time. Right now you can't even get them out, in the future after withdrawing stake is implemented, the locked out time is something like 12 months or more (I think).

You're correct which is why I don't stake on ETH. Until the powers that be decided on the rules I won't participate. Until then most of my coins are in PoW.

Why not Cardano then? The PoS consensus there is completely liquid.

In a hard fork, all rules can be changed.

That's equally true for PoW. Get 51% to make the change and your altered chain will even have the most accumulated difficulty.