A new derivation of the mass-flux equations for multicomponent diffusion in polymeric liquids is given. A precursor of the Fokker-Planck equation for a single polymer molecule is used as the starting point for the development. From this equation we derive the equation of motion for one polymer species. Then it is shown what assumptions may be made for the self-correlation tensors in order to obtain relations of the form of the Maxwell-Stefan equations for multicomponent diffusion. These results, valid for any bead-spring model, interrelate the mass-flux vectors and the stress tensor. Finally, the results are compared with the Chapman-Enskog solution of the Boltzmann equation for the limiting case of an ideal gas mixture of rigid spheres (for which no assumptions need to be made for the self-correlation tensors).