We consider evolving games with finite number of players, in which each player interacts with other randomly selected players. The types and actions of each player in an interaction together determine the instantaneous payoff for all involved players. They also determine the rate of transition between type-actions. We provide a rigorous derivation of the asymptotic behavior of this system as the size of the population grows. We show that the large population asymptotic of the microscopic model is equivalent to a macroscopic evolutionary game in which a local interaction is described by a single player against an evolving population profile. We derive various classes of evolutionary game dynamics. We apply these results to spatial random access games in wireless networks.