Let $R$ be a discrete valuation ring, $\{B_i\}_{i \in I}$ is an inductive system of finitely generated $R$-algebras and $B$ the direct limit of the inductive system. Let $X$ be a projective scheme, flat over $\mathrm{Spec}(R)$ and $\mathcal{F}$ be a coherent sheaf on $X \times \mathrm{Spec}(B)$. Does there exist some $i \in I$ and a coherent sheaf $\mathcal{F}_i$ on $X \times \mathrm{Spec}(B_i)$ such that $\mathcal{F}$ is the pull-back of $\mathcal{F}_i$ to $X \times \mathrm{Spec}(B)$ under the natural morphism $X \times \mathrm{Spec}(B) \to X \times \mathrm{Spec}(B_i)$ induced by the ring homomorphism $B_i \to B$?
P.S. If necessary, one can assume that the residue field of $R$ is algebraically closed and $X$ is regular.