We consider backward filtrations generated by processes coming from deterministic and probabilistic cellular automata. We prove that these filtrations are standard in the classical sense of Vershik's theory, but we also study them from another point of view that takes into account the measurepreserving action of the shift map, for which each sigma-algebra in the filtrations is invariant. This initiates what we call the dynamical classification of factor filtrations, and the examples we study show that this classification leads to different results.