Dust's trait reads as follows:

SCOURING SANDS: Smoke Cloud as FREE, but only to place 4 markers. At the end of each opponent's turn, deal 1 damage to each opposing character occupying or adjacent to a Smoke Cloud marker placed by Dust.

If I am playing a game with two opponents (let's call them Opponent A and B), does Opponent B deal damage to their characters at the end of Opponent A's turn (and vice versa) as well as the end of their own turns when they are in/adjacent to one of Dust's SC markers?

It seems like the way it's written, they would, but that also seems to go against the spirit of things, where you usually take damage at the end of your own turn. So maybe "each opposing character" is referencing only the characters associated with the opponent whose turn is ending?

1st off, remember that powers are written for 1 on 1 battles, so a battle royale may not translate as well.

BUT with this being said, You are reading it right, A character is either friendly or opposing, so if it's not your turn, it kicks in. Yes it kinda goes against the spirit of the game

usr3773a is correct, it activates with every opposing player's turn.

This is why all the opponents usually agree to kill Dust first when she shows up in 3+ player games.

It absolutely works that way. Dust in battle royals is... something.