The theory of Tahir-Kheli and Elliott (TKE), for analyzing correlated random walks on uniform lattices with an arbitrary concentration of background particles, is extended to treat two-sublattice systems with a potential difference between the sublattices resulting in site-dependent hopping rates. We consider first systems with overall simple-cubic and body-centered-cubic structures, along with their two-dimensional analogs. Next, two face-centered-cubic sublattices with diamond symmetry and two triangular networks forming a honeycomb lattice are studied. The tracer diffusion correlation factor and the correlation effects at general k and \ensuremath{\omega}, leading to the incoherent scattering cross section, are calculated for these systems. The limiting cases where the potential difference between the sublattices is vanishingly small lead to the earlier TKE results for a uniform lattice.