Programmable valves and actuators are widely used in man-made systems to provide sophisticated control of fluid flows. In nature, however, this process is frequently achieved using passive soft materials. Here we study how elastic deformations of cylindrical pores embedded in a flexible membrane enable passive flow control. We develop biomimetic valves with variable pore radius, membrane radius, and thickness. Our experiments reveal a mechanism where small deformations bend the membrane and constrict the pore - thus reducing flow - while larger deformations stretch the membrane, expand the pore, and enhance flow. We develop a theory capturing this highly nonmonotonic behavior, and validate the scaling across a broad range of material and geometric parameters. Our results suggest that intercompartmental flow control in living systems can be encoded entirely in the physical attributes of soft materials. Moreover, this design could enable autonomous flow control in man-made systems.