Záleží u jak které hry, u některých to jde, u některých ne.

Boom Beach:

First make sure that your game on your original device is connected to a unique Game Center account. To do so, go to your device settings > scroll down to Game Center > make sure that you are either logged in with your Apple ID or create a new account. Warning: do not share accounts with anyone – you always need to connect your account to your own Game Center account.
Then go back to your home screen, open the Game Center App, choose “games” from the bottom line, choose “Boom Beach” and then tap the three little dots in the right corner and choose “play”. When you see the “welcome back, _yourname_” banner at the top of the screen, you know you have connected your Boom Beach game to Game Center successfully.
After you have done all this you can go to your new / other device. Launch Game Center and login with the same Game Center Account you used on your old device. When opening the game you should see a pop up that asks you if you wish to load the game from your old device. Confirm and you are ready to go!