Lipschitz transport maps allow to transfer functional and isoperimetric inequalities, such as logarithmic Sobolev inequalities, from one probability measure to another. For uniformly log-concave measures on Euclidean spaces, this can be done using quadratic optimal transport, thanks to Caffarelli's contraction theorem. In this talk, I will discuss a stochastic construction of Lipschitz transport maps due to Kim and Milman, and how it works in other settings, including some non-log-concave measures and Riemannian manifolds. Joint work with D. Mikulincer and Y. Shenfeld.