You can't import bitcoin into a wallet at all. You can import 1) wallet backups and 2) private keys.
The private key (out of the public-/private- keypair created by your wallet) allows you to sign transactions 'from the corresponding address'.
The only way to 'change a private key' - in terms of controlling your coins - is to simply send your BTC's to a new address you have the private key of.
Quite a few wallets have an in-built function for that (sweeping).