This paper presents a new approach to the classical problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics. Our approach relies on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs. The first contribution exploits the dynamic formulation of Wasserstein distance, for short referred to as Wasserstein dynamics, in order to establish PCRs under dominated Bayesian statistical models. As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both stating and proving PCRs, and it sets forth a natural connection between PCRs and three well-known classical problems in statistics and probability theory: the speed of mean Glivenko-Cantelli convergence, the estimation of weighted Poincaré-Wirtinger constants and Sanov large deviation principle for Wasserstein distance. The second contribution combines the use of Wasserstein distance with a suitable sieve construction to establish PCRs under full Bayesian nonparametric models. As a novelty with respect to existing literature of PCRs, our second result provides with the first treatment of PCRs under non-dominated Bayesian models. Applications of our results are presented for some classical Bayesian statistical models, e.g., regular parametric models, infinite-dimensional exponential families, linear regression in infinite dimension and nonparametric models under Dirichlet process priors.