…Since these cards are all played as part of a single play, would the player get to finish playing them out, or would their turn just stop? And if it just stopped, what would happen to any unplayed cards? We’ve been letting the player keep them.
A: The clear answer is that if you decide to play one of these turn-ending Actions first, of course your turn would end immediately, as you have been playing it.
However, there is NO way that any remaining cards would go back into your set-aside hand. They are never intended to go into your actual hand at all, as indicated by the requirement to set your hand aside. Any cards left unplayed when you played the turn-ending card are discarded. If you wanted to play them, you should have done it before the turn-ending card.
Brain Transference: Discard remaining cards in your temporary hand and trade places with the player of your choice. Turn ends.
Time Portal: Choose a card as described and add to your set-aside hand. Discard remaining cards in your temporary hand. Turn ends.
What Do You Want: If you choose to take a Keeper or Goal out of the discard, it goes into your set aside hand. Discard remaining cards in your temporary hand. Turn ends.
I’ll Be In My Bunk: This card does not specifically say that your turn ends immediately, but you certainly can’t continue your turn if you “Excuse yourself from the game and leave the room for a few minutes.” Discard any cards remaining in your temporary hand. Leave the room.
Swap Plays For Draws and Get On With It, while they do involve having your turn end immediately, are New Rules, not Actions, so, as you can see from this answer, things would work a bit differently:
See: Can you Get On With It or Swap Plays For Draws with your cards from an Action like Draw 3 Play 2?