Shiying’s and Caroline’s paper “Approximation properties of slice-matching operators” is available on the arXiv. In this paper we iterative slicing-and-matching schemes, in particular, the approximation power of one step of such schemes. We show an invariance and an equivariance property with respect to global rigid transformations, approximation of the target measure, and derive a connection between the Wasserstein and the sliced-Wasserstein distances.