Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, the functor $$\mathcal{F}\longmapsto (\mathcal{F}(C\setminus\{x\}),\mathcal{F}_x,\iota_x)$$ from the category of coherent sheaves of $\mathcal{O}_C$-modules to the category of triplets $(M,N,\iota)$ where $M$ is a finitely generated $\mathcal{O}_C(C\setminus\{x\})$, $N$ is a finitely generated $\mathcal{O}_{C,x}$-module and $\iota_x:M\otimes_{\mathcal{O}_C(C\setminus\{x\})}k(C)\to N\otimes_{\mathcal{O}_{C,x}}k(C)$ is an isomorphism of vector spaces over $k(C)$ (which corresponds to the gluing data in the latter case) is an equivalence of categories.

Let $R$ be a Dedekind domain which is also a $k$-algebra. I want to know whether the same machinery applies for $C_R:=C\times_k \operatorname{Spec} R$ with the subscheme $\{x\}\times_k \operatorname{Spec}R$. Namely, given a triplet $(M,N,\iota)$ where $M$ is a finitely generated $\mathcal{O}_C(C\setminus\{x\})\otimes R$-module, $N$ is a finitely generated $\mathcal{O}_{C,x}\otimes R$ and $\iota$ is a $Q:=\operatorname{Quot}(k(C)\otimes_k R)$-linear isomorphism from $M\otimes_{\mathcal{O}_{C}(C\setminus \{x\})\otimes_k R} Q$ to $N\otimes_{\mathcal{O}_{C,x}\otimes_k R} Q$, can I get a corresponding coherent sheaf on $C_R$?

Sorry for the heavy notations. Any "elementary" answer is very welcome!

Many thanks!