  We prove that the group of almost-automorphisms of the infinite rooted
regular $d$-ary tree $\mathcal{T}_d$ arises naturally as the Thompson-like
group of a so called $d$-ary cloning system. A similar phenomenon occurs for
any R\"over-Nekrashevych group $V_d(G)$, for $G\le Aut(\mathcal{T}_d)$ a
self-similar group. We use this framework to expand on work of Belk and
Matucci, who proved that the R\"over group, using the Grigorchuk group for $G$,
is of type $F_\infty$. Namely, we find some natural conditions on subgroups of
$G$ to ensure that $V_d(G)$ is of type $F_\infty$, and in particular we prove
this for all $G$ in the infinite family of \v{S}uni\'c groups. We also prove
that if $G$ is itself of type $F_\infty$ then so is $V_d(G)$, and that every
finitely generated virtually free group is self-similar, so in particular every
finitely generated virtually free group $G$ yields a type $F_\infty$
R\"over-Nekrashevych group $V_d(G)$.
