Let $\pi \colon X \to T$ be a flat projective morphism, and let $Y$ be a closed sub-scheme of $X$ which is flat over $T$. We can assume that everything is defined over the complex numbers, and $T$ is one dimensional, say smooth and affine.
I want to consider the blow-up $\eta \colon X' \to X$ with centre $Y$. I have the following three questions:
Is $X'$ flat over T?
Is $(X')_t$ the blow-up of $X_t$ at $Y_t$? Where $t$ is a closed point of $T$, and the sub-index $t$ means that we are taking the fibre over $t$.
In case of positive answer to the previous question, does the exceptional divisor $E$ of $X'$ restrict to the exceptional divisor of $(X')_t $?
I think the answer is, thanks to the flatness of $Y$, always positive; however, I am not that sure about the proofs I have, and I could not find any reference.
