We study acoustic-phonon-induced relaxation of charge excitations in single and tunnel-coupled quantum dots containing few confined interacting electrons. The full configuration interaction approach is used to account for the electron-electron repulsion. Electron-phonon interaction is accounted for through both deformation potential and piezoelectric field mechanisms. We show that electronic correlations generally reduce intradot and interdot transition rates with respect to corresponding single-electron transitions, but this effect is lessened by external magnetic fields. On the other hand, piezoelectric field scattering is found to become the dominant relaxation mechanism as the number of confined electrons increases. Previous proposals to strongly suppress electron-phonon coupling in properly designed single-electron quantum dots are shown to hold also in multielectron devices. Our results indicate that few-electron orbital degrees of freedom are more stable than single-electron ones