Do Strahd Von Zarovich legendary actions movement trigger sunlight damage?

i read this quesion about legendary actions do not trigger effects from spells that happen at the beginning of the monster turn, but since I’m playing in curse of strahd, i wondered: does strahd von zarovich, that can do legendary action and move out of its turn, can run (as a legendary action) in the sunlight with no damage ? since the sunlight do damage only when he start his turn in the area of effect (and not like moonbeam spell when he go inside it).