If you suffer persistent damage, you (usually) get a DC15 flat roller to remove it. Various things can trigger this jet and / or change the DC, but it certainly comes right after taking the persistent damage.
Damage from multiple sources only accumulates if it is of a different type. that is, I could have 2 effects of persistent fire damage 2, and on acid 1d6, and I would take 1d6 of acid, and only 2 fires.
The questions are: do you need a separate token to delete each instance of persistent damage? Or can all of this be deleted with a single successful check? And if they are of a different type? If I have two cases of persistent fire damage and one of acid, is it one, two or three rollers to get me out of the injured locker?