In this talk, I will describe recent progress on these questions. I will first present an entropy-dissipation approach that yields near-i.i.d. $N^{-1/2}$ finite-particle convergence rates in kernel Stein discrepancy, as well as convergence in Wasserstein distance and long-time propagation of chaos (POC) for time-averaged particle laws. I will then discuss a cutoff principle that combines finite-time propagation-of-chaos estimates with quantitative convergence of the particle and mean-field dynamics to the target. This gives uniform-in-time POC in kernel Stein discrepancy and Wasserstein metrics, but the associated rates are logarithmic in $N$.

Finally, I will show that for noisy SVGD, obtained by adding a small Brownian diffusion to the particle dynamics, the resulting Fisher information dissipation can be utilized to obtain polynomial-in-$N$ uniform-in-time POC rates.