Damage-over-time effects are not associated with the entity that caused them. This has various effects:

• The advancement triggers player_hurt_entity and entity_hurt_player doesn't trigger from, for example, fire aspect sword fire or wither skeleton wither damage.
• Mobs will not drop experience when killed by the fire damage of a player's fire aspect sword, if sufficient time has passed since the last tracked hit.
• Death statistics are not updated when killed by, for example, the wither effect caused by a wither's skull.
• Illager captain doesn't give bad omen in the said cases (MC-179648)

Regarding advancement triggers, fire does trigger on its initial hit, but doesn't on subsequent burning damage. Splash and lingering potions of harming do work (but poison does not). All tipped arrows do not trigger when the is_magic condition is used (should possibly be its own report, but it's close enough to include in here, I think).