This talk is about Stein optimal transport, a methodology for Bayesian inference that pushes an ensemble of particles along a predefined curve of tempered probability distributions. The driving vector field is chosen from a reproducing kernel Hilbert space and can equivalently be obtained from either a suitable kernel ridge regression formulation or as an infinitesimal optimal transport map. The update equations of Stein-OT resemble those of Stein variational gradient descent (SVGD), but introduce a time-varying score function as well as specific weights attached to the particles. I will discuss the geometric underpinnings of Stein-OT and SVGD, and Ð time permitting - connections to MCMC and the theory of large deviations.