We study the critical behavior of a driven interface in a medium with random pinning forces by analyzing spatial and temporal correlations in a lattice model recently proposed by Sneppen [Phys. Rev. Lett. 69, 3539 (1992)]. The static and dynamic behavior of the model is related to the properties of directed percolation. We show that, due to the interplay of local and global growth rules, the usual method of dynamical scaling has to be modified. We separate the local from the global part of the dynamics by defining a train of causal growth events, or ‘‘avalanche,’’ which can be ascribed a well-defined dynamical exponent zloc=1+ζc≃1.63, where ζc is the roughness exponent of the interface.