I will describe recent work with J.M. Bouclet (Lille) on the propagation of coherent states up to times logarithmic in hbar under quantized perturbed cat maps. We show that, for long enough times, the quantum evolution equidistributes the coherent states throughout phase space. The proof requires a good control on the error term in the Egorov theorem on the one hand and on the classical rate of mixing on the other. This generalizes to perturbed cat maps a result obtained previously with F. Bonechi.