We introduce the Mean-Field Actor-Critic (MFAC) flow, a continuous-time learning dynamics for solving mean-field games (MFGs), drawing on ideas from reinforcement learning, generative modeling, and optimal transport. The MFAC framework jointly evolves the actor, critic, and distribution through gradient-based updates, with the distribution governed by a novel Optimal Transport Geodesic Picard (OTGP) flow. The OTGP flow drives the distribution toward equilibrium along Wasserstein-2 geodesics. We rigorously analyze the MFAC flow using Lyapunov functionals and establish global exponential convergence under suitable time scales. The analysis highlights the coupled structure of the algorithm and offers practical guidelines for choosing learning rates. Numerical results further support the theory and demonstrate the effectiveness of the proposed approach. This is joint work with Mo Zhou (UCLA) and Haosheng Zhou (UCSB).