Biological structures exhibit a level of complexity, functionality, and hierarchy that, if fully understood at a mechanistic level, could usher in the next generation of complex designer materials. For example, biological hydrogels act as selective permeability barriers by filtering nano-scale particles based on size as well as biochemical and biophysical interactions. However, for a class of situations that includes the Nuclear Pore Complex, the mechanism of this filtering has proven challenging to untangle because large non-binding particles are caged by the surrounding polymer network while binding particles exhibit increased, not decreased, mobility. We present an equilibrium mechanism for this counter-intuitive filtering strategy that does not require energy consumption. We show that selective mobility can be achieved and controlled in a simple crosslinked polymer gel by coupling binding to crosslink dynamics. Our results lead to specific design rules for manufacturing complex selective gels and could help explain how the Nuclear Pore Complex attains selectivity.