Let $X\rightarrow S$ be a fibration of relative dimension at most two and let $(X,Δ)$ be a klt pair for which $K_X+Δ\equiv_S 0$. We show that there are only finitely many Mori chambers and Mori faces in the movable effective cone $\mathcal{M}^e(X/S)$ up to the action of relative pseudo-automorphisms of $X/S$ preserving $Δ$.