This talk is about Stein transport, a novel 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 transport 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 transport and SVGD, and - time permitting - connections to MCMC and the theory of large deviations.