Background. There is an important group-theoretic notion of growth rate, defined for finitely-generated groups $G$ equipped fixed finite generating sets $S$. The growth rate is (the equivalence class of) the function $\gamma_{G,S}(n)$ "counting" the number of vertices in the ball of radius $n$ in the Cayley graph of $G$ with respect to the generating set $S$. For instance, one of the foundational results in this direction is Gromov's theorem on groups of polynomial growth.
Much is known about functions $\gamma_{G,S}(n)$ but much remains unknown.
Question:
What is known about growth rates of sequences of finite groups?
A bit more precisely:
What is known about asymptotics of the sequences $\gamma_{G_i,S_i}(n_i)$, for various classes of sequences $(G_i)$ of finite groups satisfying $|G_i|\to\infty$?
I am especially interested in the setting when all finite groups $G_i$ are simple. What is known about growth rates in this case?