If my Mimikyu V with the "Dummy Doll" Ability protection is moved to or from my Bench during my opponent's turn by a switching action such as Boss's Orders is it no longer protected?
It is still protected. It is an effect from an Ability, not from an attack, and so is not removed by moving to or from the bench.
Source: Battle Styles FAQ; TPCi Rules Team (2021-03-18)

Relevant cards

Note: All applicable cards may not be listed below.
SWSH5_62
SWSH5_62