If the PCs can make a faction lose the requisite level of Turf, then their Tier should decrease.
Example... say the PCs go to war with Baron Carter, who is Tier 2 based on his stronghold of Adamstown and contested control over the rest of East Patterson. If they seize enough Turf, or cause him to lose enough Turf, that you think he'd be knocked down a level, then knock him down. Like, I think that if he suffered a string of defeats and was forced back to his core territory in Adamstown, he would fall to Tier 1/Local Turf, because that isn't significant enough to maintain his power at Tier 2/Region Turf.