It has recently been proved by J.M.C. Clark et al. that the relative entropy (or Kullback-Leibler information distance) between two nonlinear filters with different initial conditions is a supermartingale, hence its expectation can only decrease with time. This result was obtained for a very general model, where the unknown state and observation processes form jointly a continuous-time Markov process. The purpose of this paper is (i) to extend this result to a large class of f-divergences, including the total variation distance, the Hellinger distance, and not only the Kullback-Leibler information distance, and (ii) to consider not only robustness w.r.t. the initial condition of the filter, but also w.r.t. perturbation of the state generator. On the other hand, the model considered here is much less general, and consists of a diffusion process observed in discrete-time.