Overlimiting current (OLC) through electrolytes interfaced with perm-selective membranes has been extensively researched in recent years for understanding the fundamental mechanisms of transport and developing efficient applications from electrochemistry to sample analysis and separation. Predominant mechanisms responsible for OLC include surface conduction, convection by electro-osmotic flow, and electro-osmotic instability depending on input parameters such as surface charge and geometric constrictions. This work studies how a network of microchannels in a non-uniform array, which mimicks a natural pore configuration, can contribute to OLC. To this end, micro/nanofluidic devices are fabricated with arrays of parallel microchannels with non-uniform size distributions. All cases maintain the same surface and bulk conduction to allow probing the sensitivity only by the non-uniformity of the channels. Both experimental and theoretical current-voltage relations demonstrate that OLCs increase with increasing non-uniformity. Furthermore, the visualization of internal recirculating flows indicates that the non-uniform arrays induce flow loops across the network enhancing advective transport. These evidences confirm a new driving mechanism of OLC, inspired by natural micro/nanoporous materials with random geometric structure. Therefore, this result can advance not only the fundamental understanding of nanoelectrokinetics but also the design rule of engineering applications of electrochemical membrane.