Charge inversion and mobility of a macroion in the presence of multivalent counterions and monovalent coions under an external electric field are studied by molecular dynamics simulations. The hydrodynamic interactions are treated with the explicit use of neutral-particle solvent. In a weak electric field, a complex of the macroion with condensed counterions and coions drifts along the electric field, in the direction proving the inversion of the charge sign. Very large electric field disrupts charge inversion stripping the adsorbed counterions off the macroion, which happens at the order of macroion unscreened electric field consistently with the correlation theory of charge inversion.