  A diagrammatic technique for two-particle vertex functions is used to
describe systematically the influence of spatial quantum coherence and
backscattering effects on transport properties of noninteracting electrons in a
random potential. In analogy with many-body theory we construct parquet
equations for topologically distinct {\em nonlocal} irreducible vertex
functions into which the {\em local} one-particle propagator and two-particle
vertex of the coherent-potential approximation (CPA) enter as input. To
complete the two-particle parquet equations we use an integral form of the Ward
identity and determine the one-particle self-energy from the known irreducible
vertex. In this way a conserving approximation with (Herglotz) analytic
averaged Green functions is obtained. We use the limit of high spatial
dimensions to demonstrate how nonlocal corrections to the $d=\infty$ (CPA)
solution emerge. The general parquet construction is applied to the calculation
of vertex corrections to the electrical conductivity. With the aid of the
high-dimensional asymptotics of the nonlocal irreducible vertex in the
electron-hole scattering channel we derive a mean-field approximation for the
conductivity with vertex corrections. The impact of vertex corrections onto the
electronic transport is assessed quantitatively within the proposed mean-field
description on a binary alloy.
