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