<p>The eigenvalue spectrum of the sum of large random matrices that are mutually “free”, i.e., randomly rotated, can be obtained using the formalism of <i>R</i>-transforms, with many applications in different fields. We provide a direct interpretation of the otherwise abstract additivity property of <i>R</i>-transforms for the sum in terms of a dynamical evolution of “particles” (the eigenvalues), interacting through two-body and higher-body forces and subject to a Gaussian noise, generalizing the usual Dyson Brownian motion with Coulomb interaction. Interestingly, the appearance of an outlier outside of the bulk of the spectrum is signalled by a divergence of the “velocity” of the generalized Dyson motion. We also investigate the evolution of eigenvectors, extend our results to products of free matrices, and study the singular values of a non-Hermitian matrix perturbed by a bi-invariant random matrix.</p>