If a civilization were overrun the conquering civ could choose one of the characterisics or maybe the UU. This way each conquered civ could have a cultural impact on it's new masters. This might not always be positive either based on the government type. It could be designed to create basic disenchantment within the empire more than just resistive citizans.

How would you determine if a civ overran another? If they captured their last city - what if some other civ eliminated all the other ones?

If there's a possible negative impact, then it shouldn't be by choice.

If the goal is just to make people unhappy, then it seems redundant, since there are already conquered people will already be unhappy, and unhappiness only leads to one thing.

maybe done by who conquers the capital(usaualy the cultral epicenter of a civ)