The first part of this talk will review the use of stochastic optimization to solve problems in measure transportation. In a second part, we shall focus on the slice-matching scheme, an efficient iterative method for distribution matching based on sliced optimal transport. This corresponds to stochastic gradient descent in the space of probability distributions, with discrete step sizes. We will investigate convergence to the target distribution and derive quantitative non-asymptotic rates. (joint with Co-auteurs: Claire Boyer, Kimia Nadjahi)