  In this article we study base change of Poincar\'e series along a
quasi-complete intersection homomorphism $\varphi\colon Q \to R$, where $Q$ is
a local ring with maximal ideal $\mathfrak{m}$. In particular, we give a
precise relationship between the Poincar\'e series $\mathrm{P}^Q_M(t)$ of a
finitely generated $R$-module $M$ to $\mathrm{P}^R_M(t)$ when the kernel of
$\varphi$ is contained in $\mathfrak{m}\,\mathrm{ann}_Q(M)$. This generalizes a
classical result of Shamash for complete intersection homomorphisms. Our proof
goes through base change formulas for Poincar\'e series under the map of dg
algebras $Q\to E$, with $E$ the Koszul complex on a minimal set of generators
for the kernel of $\varphi.$
