We extend the moment propagation method to capture the combined effects of adsorption/desorption of charged tracers, their migration under local and applied electric fields, as well as their advection by the local velocity of the fluid. This is achieved by combining previous developments for the separate description of these phenomena, in particular taking advantage of the Lattice Boltzmann Electrokinetics method to capture electrokinetic effects in the underlying fluid. We validate the method on the case of dispersion by an electro-osmotic flow in a slit-pore with charged walls and counter-ions in the absence of added salt. We compute the velocity auto-correlation function of charged and neutral tracers, from which we extract their average mobility and dispersion coefficient. Analytical results for the former allow to validate the algorithm, while the latter illustrates an example of property which can be provided by the moment propagation method when no analytical results are available. For both properties, we discuss the combined effects of the surface charge, of the tracer valency and of the adsorption/desorption rates.