  We prove that pushouts $A*_CB$ of residually finite-dimensional (RFD)
$C^*$-algebras over central subalgebras are always residually
finite-dimensional provided the fibers $A_p$ and $B_p$, $p\in \mathrm{spec}~C$
are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman.
This then allows us to prove that certain central pushouts of amenable groups
have RFD group $C^*$-algebras. Along the way, we discuss the problem of when,
given a central group embedding $H\le G$, the resulting $C^*$-algebra morphism
is a continuous field: this is always the case for amenable $G$ but not in
general.
