A nonlinear 3D model for mass transport into immiscible polymeric blends is developed by explicitly incorporating the interface dynamics into the transport equations. The interface is characterized, on a mesoscopic level of description, by a scalar Q(r,t) and a second-order tensor q(r,t) respectively describing the local size and anisotropy densities of the interfacial area. The newly obtained constitutive equation for the diffusion mass flux density extends Fick's first law by involving two additional terms accounting for the local changes of the interface morphology. The model provides an expression for the distribution of both isotropic (Laplace) and anisotropic stresses created by mass transport within the immiscible polymeric blend. The governing equations are parametrized by the free energy density that includes a mixing part and an excess energy term attributed to the presence of the interface. We investigate in more detail a one-dimensional sorption process of a solvent into a thin immiscible blend consisting of a matrix and a dispersed phase. Three dimensionless groups of physical parameters arise in the ID dimensionless formulation; two are coupling constants that explicitly relate diffusion to the interface dynamic changes, and one is the diffusion Deborah number. Numerical results show that diffusion becomes non-Fickian for values of Deborah number approaching unity. The time evolution of the calculated mass uptake, swelling, stresses, and total size and anisotropy densities provides a good indication of the effects of diffusion-interface interaction on both mass transport and the morphology of the interface.