Given an asymptotically cylindrical special Lagrangian submanifold L with fixed boundary in an asymptotically cylindrical Calabi-Yau 3-fold X, we determine conditions on a decay rate γ which make the moduli space of (local) special Lagrangian deformations of L in X a smooth manifold and show that it has dimension equal to the dimension of the image of H¹_cs(L,R) in H¹(L,R) under the natural inclusion map, [ξ] → [ξ]. Then we prove the analogous result for asymptotically cylindrical special Lagrangian submanifolds with moving boundary.