In what follows, I assume all schemes are Noetherian and of finite type.
It follows from the universal property that one can calculate the blowup of a coherent ideal sheaf $\mathscr{I}$ on a scheme $X$ Zariski locally. That is, taking a Zariski neighborhood $i : U \hookrightarrow X$, we have $\text{Bl}_U(i^{-1}\mathscr{I})$ is (isomorphic to) an open subscheme of $\text{Bl}_X(\mathscr{I})$.
My question is, does the above hold étale locally? In other words, taking an étale cover of $X$, pulling back the ideal sheaf to this cover, I can compute the blowup on the cover. Will this be an étale cover for the blowup $\text{Bl}_X(\mathscr{I})$? I've tried searching but can't find anyone answering (or asking!) this question.
Copy-pasting from the comments:
Blowing-up commutes with flat base changes. So the answer is yes. The proof can found at the Stacks Project.