De Giorgi popularized that even in the Riemannian setting, gradient flows allow for a variational time discretization that only appeals to the metric, not the differential structure. The JKO scheme is an instance of these minimizing movements, when the (informal) Riemannian setting is given by the Wasserstein geometry, and the functional by the (relative) entropy. This explains why McCann's displacement convexity translates into contractivity of the diffusion equation. Bochner's formula relates the dimension and the Ricci curvature of the underlying manifold to contractive rates. In this picture, De Giorgi's inequality characterization of a gradient flow can be assimilated to a large deviation principle for particle diffusions.