This is a theoretical study on the mass transport due to partially reflected long surface waves in a two-layer viscous system, which can be closed or open at its far end. Based on Lagrangian coordinates, a perturbation analysis is carried out to the second order to find the mean Lagrangian drifts in the two layers, where the lower fluid is taken to be much more viscous than the upper one. The free-surface and interfacial set-ups are also found as part of the solutions. A single analytical expression is obtained for the mass transport velocity in each layer, incorporating all the cases where the wave can be progressive, standing or partially standing, and the domain can be closed or open so that a return current may or may not exist. Through some numerical calculations, the patterns of flow in the recirculation cells due to the standing component of the wave, and in the unidirectional drifts due to the progressive component of the wave in a closed system are shown to vary with the lower-layer fluid viscosity. It is possible that, under some specific conditions, the mass transport in the core region of the upper layer is completely quiescent despite the existence of some strong drifts in the lower layer. The mean flow structures in the two layers can also respond rather differently to a change in the reflection coefficient in the presence or the absence of the return current.