The migration of a polymer chain flowing within a confining channel under combinations of an external force and pressure-driven flow is examined using the kinetic theory of an elastic dumbbell. When acting in conjunction, the polymer migrates to the center of the channel more strongly than under the action of either the external force or pressure gradient alone. When the pressure gradient and external force act in opposite directions, the polymer can migrate towards the boundaries. However, the polymer always migrates towards the center of the channel for sufficiently high external forcings. The theory qualitatively reproduces results of recent simulations as well as experiments on the combined electrophoresis and pressure-driven flow of DNA chains. The theory is consequently modified to account for the screening of hydrodynamic interactions due to the action of the electric field on the counterions surrounding the polyelectrolyte. The analysis can qualitatively capture the experimental results if the interactions are only partially screened.