Betti cones over fibre products

Let $R$ be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded $R$-modules. As an application of this, we show that the existence of an $R$-module of finite regularity and infinite projective dimension forces $R$ to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded $R$-modules, and show that when $\operatorname{depth}(R)=1$, they are spanned by the Betti tables of pure $R$-modules if and only if $R$ is Cohen-Macaulay with minimal multiplicity.