neotenie, to claim those forked coins you need to have bitcoin in your wallet in time when the fork happens, and then that forked coins are always associated with the relevant address where you hold your bitcoin. You move bitcoin before fork coins claim for security reasons, just in case you download fake wallet or in any other way expose your private keys / seed. In this way your bitcoin is safe in new wallet, and that is most important.

That proof that you have bitcoin in time of fork is something that is on blockchain, and that information stays there even when you move bitcoin to new wallet / address.