In this talk, we consider matrix-valued processes described as solutions to stochastic differential equations of a very general form. We study the family of empirical measure-valued processes constructed from the corresponding eigenvalues. We show that this family, indexed by the size of the matrix, is tight under very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its subsequences as solutions to an integral equation. Using this result, we explore certain universality classes of random matrix flows, which generalize classical results related to Dyson Brownian motion and squared Bessel particle systems. We also identify new phenomena, such as the existence of generalized Marchenko-Pastur distributions supported on the real line. Additionally, we introduce universality classes associated with generalized geometric matrix Brownian motions and Jacobi processes. Finally, under certain conditions, we study the convergence of the empirical measure-valued process of eigenvalues associated with matrix flows to the law of a free diffusion.