This question arose from an unsuccessful attempt to settle another question of mine: Vector fields on complete intersections

Let $X\to Y$ be a smooth projective morphism of noetherian schemes and let $\mathcal{F}$ be a locally free (coherent) sheaf on $X$ such that all direct images $R^i f_* \mathcal{F}$ are free. Are there any reasonable conditions which would imply that the direct images of the twisted sheaf $\mathcal{F}(1)$ and/or of the dual $\mathcal{H}om_{\mathcal{O_X}}(\mathcal{F},\mathcal{O}_X)$ are locally free?

I can't see any such conditions, but I may be missing something. If this helps, one can assume that $Y$ is the spectrum of a discrete valuation ring.

upd: counter-examples would be welcome as well, i.e. an example of $X,Y,f,\mathcal{F}$ as above such that all direct images of $\mathcal{F}$ are locally free, but some direct image of the dual is not.