Azopolymer films exposed to nonuniform illumination exhibit a phenomenon of light-induced mass transport, leading to the formation of permanent relief patterns on the film surface. Its underlying microscopic mechanism remains unclear, despite many years of research effort. Here we introduce a model of the mass migration process based on anisotropic light-driven molecular diffusion. A key ingredient of our model is an enhanced molecular diffusion in proximity of the free polymer surface, which is essential for explaining, in particular, the recently observed spiral-shaped reliefs resulting from vortex-beam illumination.