  The aim of this article is to study semigroups of composition operators on
the BMOA-type spaces $BMOA_p$, and on their "little oh" analogues $VMOA_p$. The
spaces $BMOA_p$ were introduced by R. Zhao as part of the large family of
F(p,q,s) spaces, and are the M\"{o}bius invariant subspaces of the Dirichlet
spaces $D^p_{p-1}$. We study the maximal subspace of strong continuity,
providing a sufficient condition on the infinitesimal generator of ${\phi}$,
under which $[{\phi}_t,BMOA_p]=VMOA_p$, and a related necessary condition in
the case where the Denjoy - Wolff point of the semigroup is in $\mathbb{D}$.
Further, we characterize those semigroups, for which $[{\phi}_t,
BMOA_p]=VMOA_p$, in terms of the resolvent operator of the infinitesimal
generator of $T_t$. In addition we provide a connection between the maximal
subspace of strong continuity and the Volterra-type operators $T_g$. We
characterize the symbols g for which $T_g$ acting from $BMOA$ to $BMOA_1$ is
bounded or compact, thus extending a related result to the case $p=1$. We also
prove that for $1<p<2$ compactness of $T_g$ on $BMOA_p$ is equivalent to weak
compactness.
