  We study the diffusion on an annealed disordered lattice with a local
dynamical reorganization of bonds. We show that the typical rearrangement time
depends on the renewal rate like $t_r \sim \tau^{\alpha}$ with $\alpha \neq 1$.
This implies that the crossover time to normal diffusion in a slow
rearrangement regime shows a critical behavior at the percolation threshold.
New scaling relations for the dependence of the diffusion coefficient on the
renewal rate are obtained. The derivation of scaling exponents confirms the
crucial role of singly connected bonds in transport properties. These results
are checked by numerical simulations in two and three dimensions.
