We study by coarse-grained molecular-dynamics simulations the liquid flow in a slit channel with the inner walls coated by semiflexible polymer brushes. The distance between walls is close enough such that polymers grafted to opposing walls interact among each other and form bundles across the channel in poor solvent conditions. The solvent is simulated explicitly, including particles that fill the interior of the channel. The system is studied in equilibrium and under flow, by applying a constant body force on each particle of the system. A non-linear relation between external force and flow rate is observed, for a particular set of parameters. This non-linear response is linked to a morphological change of the polymer brushes. For large enough forces, the bundle structures formed across the channel break as the chains lean in the direction of the flow, and clear the middle of the channel. This morphological alteration of the polymer configurations translates in a sudden increase in the flow rate, acting as a pressure-responsive gate. The relation between flow and external force is investigated for various parameters, such as grafting density, quality of the solvent and polymer bending rigidity. We observe a non-monotonic dependence of the flow as a function of the polymer rigidity, and find an optimum value for the persistence length. We also find that the force threshold at which the morphological changes happen in the polymer brush, depends linearly on the grafting density. These findings can lead to new flow control techniques in micro and nano-fluidic devices.