One way is to use an encryption module on the device that, rather than storing the keys just encrypts the keys and holds an encryption key that you can’t extract, and can do various crypto operations.
Now you ask the module to do a secure key exchange algorithm with the new device, meditated by a party the module trusts, like apple or something.
Now both devices share a secret key, and they trust that the other is owned by the same user because the owner verified with apple who then signed the exchange messages.
Old device decrypts with the old key, and encrypts with the new key, never letting the data leave the secure module. Send the data to the new device which can do the reverse, and both devices forget the shared password.
Overall, minor weaknesses like storing keys in the cloud encrypted by a key derived from a password that the cloud never sees, while objective weaknesses, are still significant net improvements to security over passwords.