This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by Carmona and Lacker, we consider a fully non-Markovian setting allowing for drift control and interactions through the joint distribution of players' states and controls. We provide first a characterisation of mean-field equilibria as arising from solutions to a novel kind of McKean-Vlasov backward stochastic differential equations, for which we provide a well-posedness theory. We incidentally obtain there unusual existence and uniqueness results for mean-field equilibria, which do not require short time horizon, separability assumptions on the coefficients, nor Lasry and Lions's monotonicity conditions, but rather smallness, or alternatively regularity, conditions on the terminal reward and a dissipativity condition on the drift. We then take advantage of this characterisation to provide non-asymptotic rates of convergence for the value functions and the Nash-equilibria of the N-player version to their mean-field counterparts, for general open-loop equilibria. An appropriate reformulation of our approach also allows us to treat closed-loop equilibria, and to obtain convergence results for the master equation associated to the problem.