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Let $X$ and $Y$ be smooth projective varieties over a field $k$ admittingmorphisms $f:X \to T$ and $g:Y \to T$ to a third variety $T$. We formulateconditions on a derived equivalence $\Phi:D(X) \to D(Y)$ ensuring that $\Phi$is induced by a complex $P \in D(X \times_T Y )$, defining derived equivalencesbetween the fibers of $f$ and $g$. We apply our results to the canonicalfibration and albanese fibration.

Derived equivalences over base schemes and support of complexes