The electrodiffusiophoresis of a large-zeta-potential (ζ) particle in weak fields is investigated. In this large-ζ regime, Debye-layer kinetics determines O(1) perturbations to the electric- and concentration fields in the surrounding electroneutral solution. Taking these effects into account, the expressions of the slip-flow coefficient and the effective surface boundary-conditions for the electric- and concentration fields are derived. For binary and symmetric electrolyte where only one ion species carries the current in the electroneutral domain, the far-field salt gradient as related to the electric field is determined. The electrodiffusiophoretic mobility is obtained for three particle geometries: sphere, cylinder and spheroid arbitrarily oriented with respect to the externally applied field. Strong departure from Smoluchowskian behavior is found. If co-ion is the current carrier, the mobility is independent of ζ, regardless of the body shape. Also, the hydrodynamic flow-field is irrotational. If counter-ion is the current carrier, the problem formulated in terms of a properly-defined scalar field (Ω), which embodies both the electric potential (Ψ) and the salt concentration, becomes formally identical to the one addressed in our previous work, concerning the small-ζ regime, with negligible salt gradients. Then, all the results obtained in that study are extended and applied even to the large-ζ regime considered here, provided the new expressions now derived for the surface boundary conditions and the slip-flow coefficient are employed and Ω is used in place of Ψ. The present results are discussed also in comparison with the classical studies of Dukhin et al and O'Brien et al concerning electrophoresis of highly charged particles with no salt gradient at infinity, and with recent studies of electrodiffusiophoresis, which, however, neglected the fields perturbations caused by Debye-layer kinetics. It is found that the effects addressed and incorporated in the present study determine remarkably different mobility-versus-ζ behaviour as compared to those previous theories.