We examine pressure-driven nonequilibrium transport of linear, circular, and star polymers through a nanochannel containing a rectangular pit with full hydrodynamic interactions and thermal fluctuations. We demonstrate that with sufficiently small pressure differences, there is contour length-dependent entropic trapping of the polymer in the pit when the pit and the polymer sizes are compatible. This is due to competition between flow and chain relaxation in the pit, which leads to a nonmonotonic dependence of the polymer mobility on its size and should aid in the design of nanofiltration devices based on the polymer size and shape.