$\require{AMScd}$ Consider the following commutative diagram of integral domains:
\begin{CD} R_0 @>>> R_1\\ @VVV @VVV\\ R_2 @>>> R_3, \end{CD} where $R_3 \simeq R_1 \otimes_{R_0} R_2$ .
Let $M_i$ be a finitely generated module over $R_i$ for $i=1,2$ such that $M_3 \simeq R_3 \otimes_{R_2} M_2 \simeq R_3 \otimes_{R_1} M_1$. Under what minimal assumptions on morphisms between rings (e.g. faithful flatness etc.?) is it always possible to obtain a finitely generated $R_0\textrm{-module}$ $M_0$ such that $R_1 \otimes_{R_0} M_0 \simeq M_1$ and $R_2 \otimes_{R_0} M_0 \simeq M_2$.