We study Dynamic Games with randomly entering players, staying in the game for different lengths of time. Particularly, a class of Discrete Time Linear Quadratic (LQ) Games, involving a major player who has an infinite time horizon and a random number of minor players is considered. The number of the new minor players, entering at some instant of time, is random and it is described by a Markov chain. The problem of the characterization of a Nash equilibrium, consisting of Linear Feedback Strategies, is reformulated as a set of coupled finite and infinite horizon LQ optimal control problems for Markov Jump Linear Systems (MJLS). Sufficient conditions characterizing Nash equilibrium are then derived. The problem of Games involving a large number of minor players is then addressed using a Mean Field (MF) approach and asymptotic epsilon-Nash equilibrium results are derived. Sufficient conditions for the existence of a MF Nash equilibrium are finally stated.