We analyze Elman-type recurrent neural networks (RNNs) and their training in the mean-field regime. Specifically, we show convergence of gradient descent training dynamics of the RNN to the corresponding mean-field formulation in the large width limit. We also show that the fixed points of the limiting infinite-width dynamics are globally optimal, under some assumptions on the initialization of the weights. Our results establish optimality for feature-learning with wide RNNs in the mean-field regime.