We consider the random walk Metropolis algorithm on $\R^n$ with Gaussian proposals, and when the target probability is the $n$-fold product of a one dimensional law. In the limit $n \to \infty$, it is well-known that, when the variance of the proposal scales inversely proportional to the dimension $n$ whereas time is accelerated by the factor $n$, a diffusive limit is obtained for each component of the Markov chain if this chain starts at equilibrium. This paper extends this result when the initial distribution is not the target probability measure. Remarking that the interaction between the components of the chain due to the common acceptance/rejection of the proposed moves is of mean-field type, we obtain a propagation of chaos result under the same scaling as in the stationary case. This proves that, in terms of the dimension $n$, the same scaling holds for the transient phase of the Metropolis-Hastings algorithm as near stationarity. The diffusive and mean-field limit of each component is a nonlinear diffusion process in the sense of McKean. This opens the route to new investigations of the optimal choice for the variance of the proposal distribution in order to accelerate convergence to equilibrium.