We study a class of mean field stochastic games in discrete time and continuous state space. Each player has its own individual state evolution described by a stochastic difference equation which depends not only on the control of the corresponding player but also on the states of the other players. Considering the specific structure of aggregate drift and diffusion terms, we use classical asymptotic indistinguishability properties to prove a mean field convergence in distribution. The methodology is extended to multiple classes of players, each class satisfying the asymptotic indistinguishability property, and a propagation of chaos result is obtained over the hull trajectory. Finally, we derive combined backward-forward equations that characterize the mean field equilibria for finite horizon problems.