Polymers and Brownian rods have been predicted and observed to migrate across streamlines in flowing systems, potentially impacting rheological measurements, material processing, and microfluidic systems. In particular, gradients in cross-stream diffusivity give rise to concentration gradients across streamlines, in direct contrast with naive expectations from equilibrium statistical mechanics. Here, we provide a simple, physicially intuitive understanding for the subtle mechanisms that underlie this counter-intuitive effect. Specifically, we identify three minimal ingredients: (1) the cross-stream diffusivity of the solute must depend on its internal degrees of freedom, (2) internal degrees of freedom must be driven nonconservatively in a position-dependent manner, and (3) a mechanism must exist for the concentration to relax to a steady state spatial proflie. Significantly, we argue that the inhomogeneous steady-state distributions that have been observed do not result from directed cross-stream migration. In fact, we show that no migration occurs in systems without spatial relaxation. Rather, concentration gradients are established due to anisotropic rates of spatial relaxation, and the lack of directed cross-stream migration that would be found in a conservative system. We demonstrate with simple model systems analogous to Brownian rods, externally triggerable two-state molecules, and in externally imposed temperature or solute gradients, which affect steady concentration profiles beyond what would be expected from thermophoresis or diffusiophoresis. Our results have implications for separation strategies and for various microfluidic and processing flows. © 2014 American Institute of Chemical Engineers.