Inspired by the high water permeability and excellent salt rejection of hourglass-shape aquaporin channels, asymmetric channels have attracted increasing attention in recent years. In this work, we use molecular dynamics simulations to explore the transport of an ionic solution through asymmetric graphene channels under a pressure difference. We observe an interesting asymmetric desalination phenomenon when changing the channel geometry. Specifically, the fluxes of water and ions in the convergent direction are larger than those in the divergent direction; however, the salt rejection rates exhibit an opposite behavior. This is because the strong steric effect of the channel tip results in a direct ion rejection. Moreover, in the convergent transport direction, the ions can deposit inside the channel near the tip, resulting in an ion blockage effect that ultimately leads to an abnormal descending order of fluxes of water and ions with the channel size. The ion density peaks near the tip and the asymmetric occupancy distribution provide a direct validation for the ion blockage phenomenon. We also observed flux rectification for both water and ions, depending on the pressure and channel size. These results provide a fundamental understanding of the unique ionic transport in asymmetric graphene channels, which should have great implications for designing desalination devices.