  If $G$ is a finitely generated group and $X$ a $G$-set, the growth of the
action of $G$ on $X$ is the function that measures the largest cardinality of a
ball of radius $n$ in the Schreier graph $\Gamma(G,X)$. In this note we
consider the following stability problem: if $G,H$ are finitely generated
groups admitting a faithful action of growth bounded above by a function $f$,
does the free product $G \ast H$ also admit a faithful action of growth bounded
above by $f$? We show that the answer is positive under additional assumptions,
and negative in general. In the negative direction, our counter-examples are
obtained with $G$ either the commutator subgroup of the topological full group
of a minimal and expansive homeomorphism of the Cantor space; or $G$ a Houghton
group. In both cases, the group $G$ admits a faithful action of linear growth,
and we show that $G\ast H$ admits no faithful action of subquadratic growth
provided $H$ is non-trivial. In the positive direction, we describe a class of
groups that admit actions of linear growth and is closed under free products
and exhibit examples within this class, among which the Grigorchuk group.
