Firstly I'd want to see them hash the whole blockchain (not just the last block) with the post-quantum algo to make sure history is intact.

But as far as moving balances - it's up to the owners. It would start with anybody holding a balance high enough to make it worth the amount of money it would take to crack a single key. That cracking price will go down, and the value of BTC may go up. People can move over time as they see fit.