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Having taken a look at the army book carefully, read up on the FAQs, and checked around on the internet at various spots, I want to get something clear about Bestigors and their 'Despoilers' rule. The book mentions that if the unit defeats an enemy or is part of a combat in which the enemy is defeated/breaks, the unit of Bestigors "seize" any banner the enemy unit was holding, giving them a permanent +1 combat resolution bonus until they lose combat. Does this also literally take the banner away from the enemy unit, removing its +1 combat res bonus as well as any magical effects if it was a magic banner?

You just get a +1 to combat rezuntil you lose a combat, and the enemy unit loses their banner, thats it.

If a unit was completely wiped out in CC, the banner bearers would already be dead. If the enemy unit breaks in CC, all banners are automatically removed as well.

Basically yes, as there isn't a scenario I can think of where a unit breaks from CC but doesn't lose its banners.