  Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential
equations interacting through a random matrix $\mathbf J = (J_{ij})$ with
independent (not necessarily identically distributed) random coefficients. We
show that the trajectories of averaged observables of $(X_i(t))$, initialized
from some $\mu$ independent of $\mathbf J$, are universal, i.e., only depend on
the choice of the distribution $\mathbf{J}$ through its first and second
moments (assuming e.g., sub-exponential tails). We take a general combinatorial
approach to proving universality for dynamical systems with random
coefficients, combining a stochastic Taylor expansion with a moment
matching-type argument. Concrete settings for which our results imply
universality include aging in the spherical SK spin glass, and Langevin
dynamics and gradient flows for symmetric and asymmetric Hopfield networks.
