The Caffarelli contraction theorem states that optimal transport maps (for the quadratic cost) from a Gaussian measure onto measures that satisfy certain convexity properties are globally Lipschitz, with a dimension-free estimate. It has found many applications in probability, such as concentration and functional inequalities. In this talk, I will present an alternative proof, using entropic interpolation and variational arguments. Joint work with Nathael Gozlan and Maxime Prod'homme.