Many models of dynamical systems have causal interpretations that support reasoning about the consequences of interventions, suitably defined. Furthermore, local independence has been suggested as a useful independence concept for stochastic dynamical systems. There is, however, no well-developed theoretical framework for causal learning based on this notion of independence. We study independence models induced by directed graphs (DGs) and provide abstract graphoid properties that guarantee that an independence model has the global Markov property w.r.t. a DG. We apply these results to Itô diffusions and event processes. For a partially observed system, directed mixed graphs (DMGs) represent the marginalized local independence model, and we develop, under a faithfulness assumption, a sound and complete learning algorithm of the directed mixed equivalence graph (DMEG) as a summary of all Markov equivalent DMGs.