The effect of disordered interfaces and bulk impurities on perpendicular transport in metallic multilayers is considered in an effective-mass semiclassical approximation. The transmission matrix is obtained by diagrammatic perturbation theory in terms of the effective-mass and conduction-band profiles at the interface. In the weak-scattering limit specular and diffuse scattering give equally important contributions to the conductance. Predictions for the transport properties of interfaces with low concentrations of strongly scattering defects should be accessible to verification by experiments. The transition from fully ballistic (Sharvin) to diffuse transport (Drude) is described analytically both in two- and three-dimensional systems, where the former case is of relevance for transport in the two-dimensional electron gas in semiconductor heterostructures. The theory is applied to the spin-valve effect in magnetic multilayers. The magnetoconductance is described by a simple formula in terms of the mean free path for the majority- and minority-spin electrons.