Channel-mediated transport is ubiquitous in biology. A series of works by different theoreticians have sought to determine how the diffusive flux through a channel depends on (a) stochastic gating, (b) channel geometry, and (c) heterogeneous diffusion. In this paper, we derive an explicit estimate for the diffusive flux through a channel that accounts for these three factors. We show that our estimate is exact in certain parameter regimes. We further use stochastic simulations to confirm that our estimate remains accurate across a very broad range of parameters. Our estimate differs from some results in the physics literature.