For a flat morphism $f:X \rightarrow B$ and a sub scheme $Z$ of $B$ we know that the strict and total transforms of $X$ with respect to the blow up at $Z$ agree.

I want to know what happens when $f$ fails to be flat. Suppose that $f$ fails to be flat at some point $P$ and we blow up $P$, could the strict transform of $X$ agree with the total transform? It feels as if the non-flat locus of the total transform of $f$ should contain the entire exceptional divisor which in many cases would show that this is not the case. My first question is suppose we have $f:X \rightarrow B$ a morphism of varieties not flat at $P$ in $B$ and we blow up $P$ can the product $X \times_B Bl_P B$ be flat over $Bl_P B$?

In general suppose that we have a non flat morphism of varieties $f:X \rightarrow Y$ and a proper surjective morphism $p:Z \rightarrow Y$ could the product $Z\times_Y X$ ever be flat over $Z$?

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