  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 $\Phi:D(X) \to D(Y)$ ensuring that $\Phi$
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.
