Residual finiteness for central pushouts

We prove that pushouts A ∗_C B of residually finite-dimensional (RFD) C^∗-algebras over central subalgebras are always residually finite-dimensional provided the fibers A_p and B_p, p ∈ 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 ≤ G, the resulting C^∗-algebra morphism is a continuous field: this is always the case for amenable G but not in general.