In one of the upcoming Mystery Cards, it has an effect that reads:

MC006 - Case: Murder In The City Enduring
CLUE EFFECT: Lantern Corps - During your turn, when a friendly character with a listed keyword has one or more action tokens removed, after resolutions gain 1 Clue token.

So I guess the first question to ask is would Willpower/Leadership/Clearing of tokens trigger this effect? Does stuff triggering during Beginning or End of turn count as "during your turn"?

Or is the intent for the Mystery Card to only trigger during the Action Phase of a turn?

Beginning of turn and clear phase are still part of your turn. I don’t see why everything you listed wouldn’t trigger the card.