Surface diffusion is known to be of prime importance in the growth of semiconductor nanowires. In this work, we used ZnMgO layers as markers to analyze the growth mechanisms and kinetics during the deposition of ZnMgO/ZnO multilayered shells by molecular beam epitaxy on previously grown ZnO nanowire cores (so called core-shell heterostructures). Specifically, the influence of the O2 flow sent into the plasma cell on the adatom surface mobility was investigated. By carefully measuring the growth rate on the lateral facets as well as on the top of the nanowires, it is concluded that the surface diffusion length of adatoms, within the used MBE growth conditions, is very low. Such poor surface mobility explains why so few works can be found related to the spontaneous growth (without catalyst) of ZnO nanowires by MBE, contrary to other deposition techniques.