We introduce a theoretical and numerical method to investigate the flow of charged fluid mixtures under extreme confinement. We model the electrolyte solution as a ternary mixture comprising two ionic species of opposite charge and a third uncharged component. The microscopy approach is based on kinetic theory and is fully self-consistent. It allows us to determine configurational properties, such as layering near the confining walls and the flow properties. We show that, under the appropriate assumptions, the approach reproduces the phenomenological equations used to describe electrokinetic phenomena, without requiring the introduction of constitutive equations to determine the fluxes. Moreover, we model channels of arbitrary shape and nanometric roughness, features that have important repercussions on the transport properties of these systems. Numerical simulations are obtained by solving the evolution dynamics of the one-particle phase-space distributions of each species by means of a lattice Boltzmann method for flows in straight and wedged channels. Results are presented for the microscopic density, the velocity profiles, and the volumetric and charge flow rates. Strong departures from electroneutrality are shown to appear at the molecular level.