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 gets divided between the new pieces as you choose (limited by the amount of Health each of the new pieces has). Similarly, if the piece had Strife tokens attached to it, the Strife tokens get divided between the new pieces as you choose.