Is this even "new" information? Idk. But people asked on discord, so here we go:
You're under Gatekeeper's attack. You Silver Mine a Cage to your hand. That lets you play a Falconer and gain a Dungeon. You play the Dungeon with Innovation and discard 2 cards. Does Gatekeeper exile the Cage?
Now if you discard the Cage, then for sure Gatekeeper is blocked there. But the tricky part is that the other players can't see what you discarded. So if you discard the gained Cage and a Copper, no one will know anything about where Cage is. So since no one can verify where it is, the Gatekeeper shouldn't be able to exile it.
But wait, what if you didn't discard the Cage, and left it in your hand? The Cage didn't move, so Gatekeeper should exile the Cage, right? But again, we don't know what Dungeon discarded, so who can say where the Cage actually is? Because of that, I think we have to say that the Cage is lost. So instead of moving a gained card to invoke the stop moving rule, we moved a different card and invoked that rule. So that's funny.
Also what if instead of Innovating Dungeon, you Innovated Oasis and discard a Copper? In that case, we know what got discarded, there's no ambiguity where the Cage is. So I think here, Cage should get exiled.
So tldr: If you gain a card to your hand, but then something moves card(s) from your hand - and we don't know what got moved - then we have to say, the gained card is lost. At least that's my suggestion.
EDIT: Donald X. has (tentatively) stated that, if a Cage isn't discarded from your hand, then "you" know where it is, so Gatekeeper can still exile it, and the other players will just have to trust you to be honest. Feel free to argue against it!