Passing the torch
Whenever player character dies or otherwise becomes incapacitated indefinitely, the responsibility of finishing the task is passed on to another character.
Alternate name: Passing the mantle
The first video game about Passing the torch was released on November 2008.
This does not require player has limited number of lives, only that the death/loss of the previous player character is treated like this.
Not for instances where this happens as per plot or when player can freely swap between characters.