While phenomenological expressions have been given in the past for polymer flux due to gradients in polymer stress or gradients in velocity gradient, we here provide a systematic perturbation theory for the coupled equations for polymer rheology, fluid dynamics, and polymer mass transport. Analytical results are obtained for the steady-state concentration of dilute dumbbells in the absence of hydrodynamic interactions, as a function of the Weissenberg number Wi, Peclet number Pe, and the "gradient number" Gd, where the latter is the ratio of polymer equilibrium size to the characteristic distance over which velocity gradients change. At lowest order in all three perturbation variables, the theory yields a Poisson's equation with the driving term given by 2 (c) over barD tau(2) vjk,(j) v(j,j,k), where (c) over bar is the average concentration, D is the dumbbell diffusivity, tau is the relaxation time, and v is velocity with Einstein notation for the subscripts. We find using Brownian dynamics (BD) simulations in a simple Taylor vortex flow that this formula becomes accurate for small values of Gd, Wi, and Pe. For small Gd and Wi, but large Pe, we obtain accurate solutions from the Poisson's equation augmented with a simple convection term. For gradients beyond second order in Gd, continuum equations lose utility, but mesoscopic methods such as BD can then be used to determine polymer migration. We give a hierarchy of approximations that shows which equations and methods are required as the Peclet number Pe, the Weissenberg number Wi, and/or the gradient number Gd, exceed unity or as the polymer becomes concentrated or hydrodynamic interactions become important. (C) 2016 The Society of Rheology.