We study the diffusion (and conductivity) associated with the random walk of noninteracting particles on a disordered lattice characterized by bond disorder, temporal rearrangement, and spatial correlations. This paper extends previous works on dynamic bond percolation processes to situations where spatial correlations in the rearrangement process are important. Many bond effective-medium theory is used to obtain the effective diffusion coefficient Deff (ω) in such systems. The resulting Deff (ω) depends on the frequency through combinations of the form ω-i/τ, where τj are characteristic relaxation times associated with the rearrangement process. We analyze in detail a model combining single bond renewal with a two bond exchange process. The resulting DC (τ = 0) diffusion coefficient shows a new percolation threshold for the bond exchange model (in the absence of single bond renewal which eliminates the threshold altogether), and a crossover between the different limiting behaviors is seen as the different kinds of renewal process are switched on and off. Implications for ionic transport in polymeric ionic conductors are discussed.