We derive from kinetic theory, fluid mechanics and thermodynamics the minimal continuum-level equations governing the flow of a binary, non-electrolytic mixture in an isotropic porous medium with osmotic effects. For dilute mixtures, these equations are linear and in this limit provide a theoretical basis for the widely used semi-empirical relations of Kedem & Katchalsky (Kedem & Katchalsky 1958 Biochim. Biophys. Acta 27 , 229–246 ( doi:10.1016/0006-3002(58)90330-5 ), which have hitherto been validated experimentally but not theoretically. The above linearity between the fluxes and the driving forces breaks down for concentrated or non-ideal mixtures, for which our equations go beyond the Kedem–Katchalsky formulation. We show that the heretofore empirical solute permeability coefficient reflects the momentum transfer between the solute molecules that are rejected at a pore entrance and the solvent molecules entering the pore space; it can be related to the inefficiency of a Maxwellian demi-demon.