For example, if you deal 2 Damage to a City and then replace it with a Town, that Town is immediately destroyed.
If you replace a damaged piece with two or more pieces (as when "Dissolve the Bonds of Kinship" replaces a City with two Explorers), the damage get divided between the new pieces as you choose. Similarly, if the piece had Strife tokens attached to it, the Strife tokens get divided between the new pieces as you choose.