Let $S$ be the spectrum of a DVR with generic point $\eta$ and closed point $s$. Let $X$ be a flat, quasi-projective scheme over $S$. Let $X_s$ denote the special fiber, and let $Z \subset X_s$ be a closed subvariety.
On the one hand, I can consider $\mathrm{Bl}_Z(X_s)$, the blow-up of the special fiber $X_s$ at $Z$. On the other hand, I can also see $Z$ as a closed subscheme of $X$ via the closed embedding $X_s \hookrightarrow X$, and thus form the blow-up $\mathrm{Bl}_Z(X)$. The latter is a scheme over $S$, thus it makes sense to consider its special fiber $\mathrm{Bl}_Z(X)_s$.
What is the relation between $\mathrm{Bl}_Z(X_s)$ and $\mathrm{Bl}_Z(X)_s$?
I think that Proposition 3.12 of these notes by Toni Annala gives a natural commutative square
$$\begin{array}[ccc] \mathrm{Bl}_Z(X_s) & \rightarrow & \mathrm{Bl}_Z(X) \\ \downarrow & & \downarrow \\ X_s & \rightarrow & X \end{array}$$
and the horizontal maps are closed immersions. The commutativity of the diagram implies that the closed immersion $\mathrm{Bl}_Z(X_s) \rightarrow \mathrm{Bl}_Z(X)$ factors through the special fiber, thus $\mathrm{Bl}_Z(X_s)$ is a closed subvariety of $\mathrm{Bl}_Z(X)_s$. Is this actually an equality?