I'm sure I lost a couple prizes because of our move. The post office should forward anything mailed to your old address (you can fill out the change of address on their website or pick up a paper one at the post office), but some classes of mail won't get forwarded. If there is going to be someone moving into your prior address or you're moving from an apartment building where the office accepts package delivery, you can ask them nicely to call or forward anything that may show up from UPS/FedEx for a few months while you transition.
If you're entering stuff during the move, start updating the sites that can be updated with your new address about a week before you move, and start using your new address for new entries at the same time (It's a great way to memorize the new address).
You're not going to avoid missing every single prize, especially the ones you don't know are coming, but most of them should find their way to the right place.