It is well known that if two varieties are Fourier-Mukai partners, then there are strong constraints on them. For example, if one of them is Calabi-Yau the other one must be Calabi-Yau too. Similarly for smoothness (if one is smooth the other one must be smooth).

$\textbf{Question:}$ If $\pi: X \longrightarrow B$ is a fibration (for example elliptic fibration) over $B$, and similarly $\bar{\pi}: Y\longrightarrow \bar{B}$ another fibration over $\bar{B}$, and $D^b(X)\simeq D^b(Y)$ (also assume $\bar{B}$ and $B$ are birational). Then is it correct to say, $\pi$ is a flat morphism if and only if $\bar{\pi}$ is a flat morphism?

$\textbf{Comment:}$ What I have in mind is the case where $X$ and $Y$ are both elliptically fibered threefold, or at least genus one fibration.