We present a study of the extent to which the Sellwood & Binney radial migration of stars is affected by their vertical motion about the mid-plane. We use both controlled simulations in which only a single spiral mode is excited, as well as slightly more realistic cases with multiple spiral patterns and a bar. We find that rms angular momentum changes are reduced by vertical motion, but rather gradually, and the maximum changes are almost as large for thick disc stars as for those in a thin disc. We find that particles in simulations in which a bar forms suffer slightly larger angular momentum changes than in comparable cases with no bar, but the cumulative effect of multiple spiral events still dominates. We have determined that vertical action, and not vertical energy, is conserved on average during radial migration.