A three-dimensional mathematical model for sorption/desorption by a cylindrical polymeric matrix with dispersed drug is proposed. The model is based on a system of partial differential equations coupled with boundary conditions over a moving boundary. We assume that the penetrant diffuses into a swelling matrix and causes a deformation, which induces a stress-driven diffusion and consequently a non-Fickian mass flux. A physically sound nonlinear dependence between strain and penetrant concentration is considered and introduced in a Boltzmann integral with a kernel computed from a Maxwell--Wiechert model. Numerical simulations show how the mechanistic behavior can have a role in drug delivery design.