We study the spatiotemporal dynamics of a microfluidic system with a nonselective microfluidic channel gated by an ion-selective membrane which separates the ion flux paths of cations and anions. To preserve electroneutrality, the ionic concentration in the system is shown to converge to a specific inhomogeneous distribution with robust constant current fluxes. A circuit scaling theory that collapses measured asymptotic currents verifies that this is a generic and robust mechanism insensitive to channel geometry, ion selectivity, and electrolyte ionic strength. This first temporally stationary but spatially inhomogeneous depletion front can be used for modulating ionic current and for isotachophoretic isolation of low-mobility molecules and exosomes on small diagnostic chips for various medical applications that require robust high-throughput and integrated platforms.