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We consider conservative and gradient flows for $N$-particle Riesz energieswith mean-field scaling on the torus $\mathbb{T}^d$, for $d\geq 1$, and withthermal noise of McKean-Vlasov type. We prove global well-posedness andrelaxation to equilibrium rates for the limiting PDE. Combining theserelaxation rates with the modulated free energy of Bresch et al. and recentsharp functional inequalities of the last two named authors for variations ofRiesz modulated energies along a transport, we prove uniform-in-time mean-fieldconvergence in the gradient case with a rate which is sharp for the modulatedenergy pseudo-distance. For gradient dynamics, this completes in the periodiccase the range $d-2\leq s<d$ not addressed by previous work of the second twoauthors. We also combine our relaxation estimates with the relative entropyapproach of Jabin and Wang for so-called $\dot{W}^{-1,\infty}$ kernels, givinga proof of uniform-in-time propagation of chaos alternative to Guillin et al.

Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows