The trailing vines that make hanging plants beautiful are also what gets damaged first in a move; they catch on everything if left dangling. Coiling them up before you touch the hanger solves most of the risk.
Pro tip: Take a quick photo of the coiled vine pattern so you remember which loop goes where when you uncoil it at the new place.
Watch out: A hanging plant transported still hanging from its hook can swing and snap vines against walls or other cargo.