  Mean field games are limit models for symmetric $N$-player games with
interaction of mean field type as $N\to\infty$. The limit relation is often
understood in the sense that a solution of a mean field game allows to
construct approximate Nash equilibria for the corresponding $N$-player games.
The opposite direction is of interest, too: When do sequences of Nash
equilibria converge to solutions of an associated mean field game? In this
direction, rigorous results are mostly available for stationary problems with
ergodic costs. Here, we identify limit points of sequences of certain
approximate Nash equilibria as solutions to mean field games for problems with
It{\^o}-type dynamics and costs over a finite time horizon. Limits are studied
through weak convergence of associated normalized occupation measures and
identified using a probabilistic notion of solution for mean field games.
