I have moved house 8 times since joining postcrossing 13 years ago, once across the Atlantic.
I almost all cases, I used the (inexpensive/free) forwarding service provided by the respective mail service. But not a lot of cards ended up having to be forwarded because (as others said), I also set my account inactive about two months before moving. (Inactive means you are still allowed to send cards, but you won’t receive any.)
And last but not least, postcrossing now offers to automatically register all cards traveling to you when you change your address. That would make the forwarding service obsolete (esp. when it’s usually not cheap).
But I would always make my account inactive at least six weeks before my move to make travel times more realistic upon automatic registration (or you might end up with a card traveling to you from China in a day).