Using in situ liquid cell electron microscopy we investigate Pd growth in dilute aqueous Pd salt solutions containing Au nanoparticle seeds. Au-Pd core-shell nanostructures are formed via deposition of Pd(0), generated by the reduction of chloropalladate complexes by radicals, such as hydrated electrons (eaq(-)) induced by the electron beam in the solution. The size and shape of the Au seeds determine the morphology of the Pd shells, via preferential Pd incorporation in low-coordination sites and avoidance of extended facets. Analysis of the Pd incorporation on Au particles at different distances from a focused electron beam provides a quantitative picture of the growth process and shows that the growth is limited by the diffusion of eaq(-) in the solution.