The magnetoresistance of metallic multilayers in the current-perpendicular-to-plane (CPP) geometry is studied theoretically on an ab initio level using the tight-binding linear muffin-tin orbital method. The applied potential parameters were determined self-consistently for a given alloy composition within the coherent potential approximation (CPA). Lateral supercells with random arrangements of atoms of two types are used to represent disorder connected with interface interdiffusion and with alloying in the spacer. We distinguish ballistic and diffusive parts of transport and study their dependence on the type and on the strength of disorder. The theoretical approach is illustrated on disordered systems derived from the fcc-based Co/Cu/Co(001) trilayers that serve as the reference system. We find quite a good agreement with experimental data and with other calculations. On the other hand, our results also show the limited validity of the two-current series-resistor model and that vertex corrections to the CPA applied to the CPP transport are of great importance.