What happens if I mutate on top of a creature that I control until the end of a turn?
In this case, Cavern Whisperer won't be a valid target for mutation, because you are not the owner of Cavern Whisperer. This is per CR:

702.140a Mutate appears on some creature cards. It represents a static ability that functions while the spell with mutate is on the stack. “Mutate [cost]” means “You may pay [cost] rather than pay this spell’s mana cost. If you do, it becomes a mutating creature spell and targets a non-Human creature with the same owner as this spell.”  (...)


108.3. The owner of a card in the game is the player who started the game with it in their deck. (...)