It sounds like you may want Exercise 9.7 in Hartshorne's "Residues and Duality". I paraphrase the statement:
Exercise 9.7 (RD):
Let $f: X to B$ be a flat morphism of finite type of locally Noetherian preschemes. Then, $f^!(mathcal{O}_B)$ has a unique non-zero cohomology sheaf, which is flat over $B$, iff all the fibers of $f$ are Cohen-Macaulay schemes of the right dimension. Moreover $f^!(mathcal{O}_B)$ is isomorphic to (a shift of) an invertible sheaf (on $X$) iff all the fibers of $f$ are Gorenstein schemes of the right dimension.
In particular, this addresses the case you mention in your comment ($f$ Gorenstein), since then $f^!(mathcal{O}_Y)$ is locally free on $X$ and, since $f$ is flat, certainly flat over $B$.
[Aside: I believe I learned this reference from Brian's book "Grothendieck Duality and Base Change", which I think also contains a proof of this.]
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