In nature, liquids often circulate in channels textured with leaflets, cilia, or porous walls that deform with the flow. These soft structures are optimized to passively control flows and have inspired the design of novel microfluidic and soft-robotic devices. Yet, the relationship between the geometry of the soft structures and the flow properties remains so far poorly understood. Here, taking inspiration from the lymphatic system, we devise millifluidic valves, dressed with asymmetric soft leaflets that feature on-demand asymmetric flow properties, which we harness to induce flow rectification and pumping. We demonstrate that the functional mechanism at play involves an interplay between geometry, asymmetry, and fluid-structure interaction-induced nonlinearity. Our results open the way to a better characterization of biological leaflet malformations and to more accurate control of flows for microfluidics and soft-robotic systems.