Automorphism growth and group decompositions

Let $G$ be a finitely generated group with an automorphism $\phi \in \mathrm{A}\mathrm{u}\mathrm{t}(G)$, or an outer automorphism $\phi \in \mathrm{O}\mathrm{u}\mathrm{t}(G)$. Suppose that $G$ decomposes into simpler pieces on which the growth behaviour of $\phi$ and $\phi$ is known, particularly as a direct product, free product, or graph of groups. This article is devoted to the (often not entirely straightforward) problem of deducing information about the growth rates of $\phi$ and $\phi$ on the whole $G$.