Let $\mathcal{R}$ be a discrete valuation ring of unequal characteristic and $\mathcal{K}$ be its field of fractions. Let $X$ be a smooth proper curve over $Spec(K)$ and $\mathcal{X}$ be a smooth proper model over $Spec(R)$. The moduli of rank $n$ vector bundles on $X$, $Bun_n(X)$, is naturally a smooth algebraic stack over $Spec(\mathcal{K})$. My question is the following:
If we take the moduli of rank $n$ vector bundles on $\mathcal{X}$ then does it also form a $\textit{reasonable}$ algebraic stack? In particular does it give an smooth algebraic stack over $Spec(\mathcal{R})$ whose base change to $Spec(\mathcal{K})$ recovers $Bun_n(X)$? If it doesn't what goes wrong?
