Our development of the self-consistent mean-field (SCMF) kinetic theory for nonuniform alloys leads to the statement that kinetic correlations induced by the vacancy diffusion mechanism have a dramatic effect on nanoscale diffusion phenomena, leading to nonlinear features of the interdiffusion coefficients. Lattice rate equations of alloys including nonuniform gradients of chemical potential are derived within the Bragg-Williams statistical approximation and the third shell kinetic approximation of the SCMF theory. General driving forces including deviations of the free energy from a local equilibrium thermodynamic formulation are introduced. These deviations are related to the variation of vacancy motion due to the spatial variation of the alloy composition. During the characteristic time of atomic diffusion, multiple exchanges of the vacancy with the same atoms may happen, inducing atomic kinetic correlations that depend as well on the spatial variation of the alloy composition. As long as the diffusion driving forces are uniform, the rate equations are shown to obey in this form the Onsager formalism of thermodynamics of irreversible processes (TIP) and the TIP-based Cahn-Hilliard diffusion equation. If now the chemical potential gradients are not uniform, the continuous limit of the present SCMF kinetic equations does not coincide with the Cahn-Hilliard (CH) equation. In particular, the composition gradient and higher derivative terms depending on kinetic parameters add to the CH thermodynamic-based composition gradient term. Indeed, a diffusion equation written as a mobility multiplied by a thermodynamic formulation of the driving forces is shown to be inadequate. In the reciprocal space, the thermodynamic driving force has to be multiplied by a nonlinear function of the wave vector accounting for the variation of kinetic correlations with composition inhomogeneities. Analytical expressions of the effective interdiffusion coefficient are given for two limit behaviors of the vacancy, the latter treated as either a conservative species (fixed concentration) or a nonconservative species (time-dependent equilibrium concentration). Relying on the same vacancy diffusion model, we perform kinetic Monte Carlo simulations starting from a sinusoidal composition modulation in binary model alloys, with no interaction or nearest-neighbor interactions leading to clustering or ordering tendencies, along the  crystallographic direction of a body centered cubic (bcc) lattice. The resulting temporal variation of the modulation amplitude is compared to the corresponding SCMF equations. Qualitative and satisfying quantitative agreements systematically strengthen our theoretical conclusions. The model alloys are shown to be representative enough of some real alloys, so that one may expect these new heterogeneous correlation effects to be non-negligible in these alloys.