$\require{AMScd}$ Consider a given commutative diagram of commutative Noetherian domains:
\begin{CD} R_1 @>>> R_3\\ @AAA @AAA\\ R_0 @>>> R_2, \end{CD} where $R_0 = R_1 \cap R_2$, horizontal maps are localization with respect to the same multiplicative set inside $R_0$, and vertical maps are faithfully flat.
Let $M_i$ be a finite free module over $R_i$ for $i=1,2,3$ such that $M_3 = R_3 \otimes_{R_2} M_2 = R_3 \otimes_{R_1} M_1$. Let $M_0 = M_1 \cap M_2 \subset M_3$ be an $R_0$-module such that $M_2 \simeq R_2 \otimes_{R_0} M_0$.
Is it true that $M_0$ is a finitely generated projective module over $R_0$, or equivalently, a finitely generated flat $R_0$-module? If not, are there easy to state counterexamples?