If games support game saves through Game Center, then your games will be synced with the devices that have the same account.
But regarding the TeamLava transfers, you would have to contact TL to do the transfer for you or use their website.
I found that the storm8 app way to transfer TeamLava games to a new/different device was simpler.
Sorry I thought this thread was regarding TeamLava games. I guess it was for games in general 😱