Matching and estimating probability distributions are fundamental tasks in generative modeling applications. While optimal transport (OT) provides a natural framework for matching distributions, its large-scale computation remains challenging, especially in high dimensions. Our focus lies on an efficient iterative slice-matching approach initially introduced by Pitie et al. for transferring color statistics. The resulting sliced optimal transport is constructed by solving a sequence of one-dimensional OT problems between the sliced distributions, which are fast to compute and admit closed-form solutions. While these techniques have proven effective in various data science applications, a comprehensive understanding of their convergence properties remains lacking. Our primary aim is to establish an almost sure convergence, shedding light on the connections with stochastic gradient descent in the context of the sliced-Wasserstein distance. If time permits, we will also explore invariance, equivariance, and Lipschitz properties associated with one-step of the procedure, yielding recovery and stability results for sliced OT approximations with a single step. This talk is based on joint work with Caroline Moosmueller.