Derived equivalences over base schemes and support of complexes

Let $X$ and $Y$ be smooth projective varieties over a field $k$ admitting morphisms $f:X \to T$ and $g:Y \to T$ to a third variety $T$. We formulate conditions on a derived equivalence $Φ:D(X) \to D(Y)$ ensuring that $Φ$ is induced by a complex $P \in D(X \times_T Y )$, defining derived equivalences between the fibers of $f$ and $g$. We apply our results to the canonical fibration and albanese fibration.

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