The rewards will still come to the existing wallet after you send it because the balance for the active stake has been snapshotted in the previous epochs already. You will probably need some ADA on the existing wallet (1 is enough) later on to withdraw those rewards.
So let's say you transfer your funds from wallet 1
to wallet 2
to in epoch X
(and you are already actively receiving rewards every epoch to wallet 1
). You will receive rewards at the start of epochs X+1
, X+2
and X+3
to wallet 1
. From epoch X+4
, the rewards will start coming to wallet 2
(assuming you delegated from wallet 2
the same epoch you received the funds - epoch X
). Hopefully, I got the numbers right, but there should definitely be no delay/loss of rewards overall because of simply switching funds to a new wallet. Instead, it is just a question of which epoch will the rewards stop coming into wallet 1
and start coming into wallet 2
instead.