Smectic liquid crystals form freely-suspended, fluid films of highly uniform structure and thickness, making them ideal systems for studies of hydrodynamics in two dimensions. We have measured particle mobility and shear viscosity by direct observation of the gravitational drift of silica spheres and smectic islands included in these fluid membranes. In thick films, we observe a hydrodynamic regime dominated by lateral confinement effects, with the mobility of the inclusion determined predominantly by coupling of the fluid flow to the fixed boundaries of the film. In thin films, the mobility of inclusions is governed primarily by coupling of the fluid to the surrounding air, as predicted by Saffman-Delbrück theory.