Michael Van Biesbrouck
I assume that you are referring to upgrades with the Redeploy keyword.
This keyword only appears on upgrades. After the attached character is defeated, its controller may immediately move this upgrade to any of their other characters. The upgrade die moves to the new character card, even if it was in the dice pool.
So in both cases the die goes onto the new character card and can be rolled next time the character is activated. Sometimes you will get to use the die twice, other times you can't use it at all.