We calculate the eigenmode spectrum for collective multipole vibrations of the electrons of alkali clusters including coupling between surface and volume plasmons. We formulate the equations of motion for the collective variables (density and velocity potential) starting from the hydrodynamic approximation. We investigate the effect of the diffusibility of the valence electrons on their collective modes, considering an equilibrium density of the valence electrons with a smooth surface profile. This effect was not considered in previous work carried out by the first author, who assumed the equilibrium density of the valence electrons to be a constant that has the bulk value. The results indicate that, within the hydrodynamic model, surface spill-out effects lead to a tendency for a red-shift from the Mie frequency and a mixing of resonance modes when the size of the cluster decreases. The eigenmodes fulfil the linear energy-weighted sum rule, the inverse energy-weighted sum rule and orthogonality relations.