While I was creating a topic for my last idea, I was thinking that there could also be a way to visually show you leaving to a mini-game.
I was thinking things like taking a train or plane off to the mini-game location. The people that want to do this would simply sit down in a seat of the train or plane and ride it when it leaves. They would then be taken to a loading screen to either
a) be shown arriving to their destination and be in the lobby of the gamemode or
b) place into the gamemode like GMT