If $f\colon X \to S$ a proper flat of schemes map with $n$-dimensionnal fibers over a noetherian scheme $S$, the relative canonical sheaf $\omega_{X/S}:=H^n(f^!{\mathcal{O}_S})$ is a dualizing sheaf. I guess that this should imply what you want by a GAGA-type argument.
Indeed, being $f$ flat we have that $f^!G = f^*G \otimes f^!{\mathcal{O}_S}$ [Lipman, SLN 1960, Theorem 4.9.4] for $G \in D^b_c(X)$ (or, more generally, for $G \in D^+_{qc}(X)$). Now, $f^!{\mathcal{O}_S}$ is concentrated between degrees 0 and $-n$ by looking at the fibers of $f$ and its description via residual complexes. Then, taking $-n$-th cohomology on both sides and one obtains the desired result.
Unfortunately, I am not familiar enough with the analytic version of the story as in Ramis-Ruget-Verdier "Dualité relative en géométrie analytique complexe", but I guess the algebraic version may give you a clue for how to transpose the result to your setting. I would bet you do not need $S$ reduced as long as $f$ is flat.