When an effect reduces the Health of a piece (or it loses a bonus to Health), and it has already taken damage equal to its new health, is it destroyed immediately?

When an effect reduces the Health of a piece (or it loses a bonus to Health), and it has already taken damage equal to its new health, is it destroyed immediately?

Yes. (Branch & Claw rulebook p. 16)
Invaders and Dahan have Health and take Damage. As soon as they have taken as much Damage as their Health, they are destroyed, whether that comes from taking more Damage or having their Health reduced.
Unlike in some other games, doing Damage normally doesn't reduce the Health of the Invaders; there are two separate totals.