We study the hydrodynamic transport of knotted ring polymers through modulated channels, establishing that the transport velocity is strongly dependent on the ring topology for Peclet numbers smaller than unity. As soon as convection dominates, transport properties become insensitive to the presence and type of knots. We identify two distinct modes of transport, corresponding to the motion being led by the knotted or unknotted portions of the ring, most surprisingly without impact on separation efficiency. The modes can be selected by the channel geometry, and this could be harnessed to design nanofluidic devices for the continuous topological sorting of entangled biopolymers.