The current-perpendicular-to-plane (CPP) magnetotransport of a metallic sample sandwiched by two ideal leads is described at an ab initio level. The so-called "active" part of the system is either a trilayer consisting of two magnetic slabs of finite thickness separated by a nonmagnetic spacer or a multilayer formed by alternating magnetic and nonmagnetic layers. We use a transmission matrix formulation of the conductance based on surface Green's functions as formulated by means of the tight-binding linear muffin-tin orbital method. The formalism is extended to the case of lateral supercells with random arrangements of atoms of two types, which in turn allows to deal with specular and diffusive scattering on equal footing, and which is applicable also to the case of noncollinear alignments of the magnetization in the layers. Applications refer to fcc-based Co/Cu/Co(001) trilayers and multilayers, considering in detail the effect of substitutional alloying in the spacer and in the magnetic layers, as well as interdiffusion at the interfaces.