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
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.