We investigate the transport of macromolecules with different architectures through polymer brush-covered microchannels by means of molecular dynamics simulations. Linear macromolecules and star polymers with different functionalities and arm lengths are driven through the channel by a Poiseuille flow with various pressure gradients. A comparison between the diffusive dynamics in equilibrium and upon Poiseuille flows reveals dramatic changes of the transport properties. While the self-diffusion for linear chains and stars with large functionality and short arms (the latter representing the limit of hard spheres) is faster than for stars with intermediate softness, the current density of macromolecules in Poiseuille flows depends strongly on the flow strength and tends to decrease monotonically with increasing functionality. Our measurements of density and flow profiles as well as binary interactions among different species (brushes, solvent, macromolecular inclusions) shed light on complex translocation processes, which are of importance in biology and micromechanical applications.