The branching rule for the tower of wreath products of a finite group by the symmetric groups induces a stochastic process on the set of multiple Young diagrams through random transitions of boxes of the diagrams between one another. We observe dynamical multiple averaged limit shapes resulting from appropriate scaling limits, either diffusive or non-diffusive. We describe time evolution of the macroscopic multiple averaged limit shapes in terms of Voiculescu's $R$-transforms and free Lévy measures of corresponding Kerov transition measures. Our microscopic dynamics admits non-exponential pausing time distributions.