If Cast Out leaves the battlefield before its triggered ability resolves, the target permanent won't be exiled.

Auras attached to the exiled permanent will be put into their owners' graveyards. Any Equipment will become unattached and remain on the battlefield. Any counters on the exiled permanent will cease to exist.

If a token is exiled this way, it will cease to exist and won't return to the battlefield.

The exiled card returns to the battlefield immediately after Cast Out leaves the battlefield. Nothing happens between the two events, including state-based actions.

In a multiplayer game, if Cast Out's owner leaves the game, the exiled card will return to the battlefield. Because the one-shot effect that returns the card isn't an ability that goes on the stack, it won't cease to exist along with the leaving player's spells and abilities on the stack.