  We provide a proof of mean-field convergence of first-order dissipative or
conservative dynamics of particles with Riesz-type singular interaction (the
model interaction is an inverse power $s$ of the distance for any $0<s<d$) when
assuming a certain regularity of the solutions to the limiting evolution
equations. It relies on a modulated-energy approach, as introduced in previous
works where it was restricted to the Coulomb and super-Coulombic cases. The
method is also capable of incorporating multiplicative noise of transport type
into the dynamics. It relies in extending functional inequalities of
arXiv:1803.08345, arXiv:2011.12180, arXiv:2003.11704 to more general
interactions, via a new, robust proof that exploits a certain commutator
structure.
