Specialization of finite correspondence

Question

Suppose we have a finite correspondence $C\subset X\times X$, where $X/k$ is a variety. (i.e. $C$ is a closed subset of $X\times X$, such that the projection maps to $X$ are finite). Let $\mathcal{X}$ be an integral model of $X$ over $Spec R$, where $R$ is a DVR with fraction field $k$. Consider the correspondence $\mathcal{C}$ given by the closure of $C$ in $\mathcal{X}\times_{Spec R}\mathcal{X}$. Specialize $\mathcal{C}$ to the special fiber $X_s\times X_s$, to obtain a correspondence $C_s$. Is this correspondence necessarily finite? (i.e are the projection maps $\pi_{1,s},\pi_{2,s}$ to $X_s$ finite?) What if we also assumed that the maps $\pi_{1,s},\pi_{2,s}$ were proper?