We consider a class of time-varying mean-field games with denumerable state space and possibly unbounded costs. The mean-field game state’s process evolves according to a time-varying transition probability \(p^{n}\) which depends on a probability measure modeling the collective behavior of a large number of players. Considering a discounted optimality criterion and assuming that \(p^{n}\) converges suitably to a transition probability \( p^{\infty }\) , our objective is to prove the existence of a stationary mean-field equilibrium for the corresponding limiting mean-field game. Furthermore, we show that \(\left \{ p^{n}\right \} \) defines a sequence of mean-field equilibria that approximate the one corresponding to \(p^{\infty }\) .