We introduce the concept of locally acting distributed players into sequential stochastic games with general compact state and action spaces. The state transition function for the system is of local structure as well and this results in Markov properties in space and time for the describing processes. We prove that we can reduce optimality problems for local strategies to only considering Markov strategies. We further prove the existence of optimal strategies and of a value for the game with respect to the asymptotic average reward criterion.