  We consider matrix-valued processes described as solutions to stochastic
differential equations of very general form. We study the family of the
empirical measure-valued processes constructed from the corresponding
eigenvalues. We show that the 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. We use this result to study some universality classes of
random matrix flows. These generalize the classical results related to Dyson
Brownian motion and squared Bessel particle systems. We study some new
phenomenons as the existence of the generalized Marchenko-Pastur distributions
supported on the real line. We also introduce universality classes related to
generalized geometric matrix Brownian motions and Jacobi processes. Finally we
study, under some conditions, the convergence of the empirical measure-valued
process of eigenvalues associated to matrix flows to the law of a free
diffusion.
