Let $f: X \to Y$ and $g: Y \to Z$ be morphisms of schemes* such that f is flat and finite, g is proper and $R^{> 1}g_*E=0$ for all sheaves and Z is affine.

Let E be a vector bundle on Y such that $R^1 g_* E=0$. Can we say anything about $H^1 (X,f^* E)$? By the projection formula this is the same as $R^1 g_* ( E \otimes f_* \mathcal{O} ) $ and as f is flat and finite $f_* \mathcal{O}_X $ is a vector bundle. But I can't seem to be able to say much else.

[*there are many hypotheses I'd be happy to make: everything is finite type over a field, the field is algebraically closed of characteristic zero, all the schemes involved are integral, Y is regular, g is actually the restriction of a morphism of projective varieties $g': Y' \to Z'$ to an open affine patch $Z$ of $Z'$. Finally the dual of $E$ is globally generated.]

EDIT: silly me, I forgot to add some other assumptions (but thanks for the answer, I fear it could it be tweaked to still give a counterexample). $X,Y,Z$ have all the same dimension (equal to 3), g is birational, $Rg_* \mathcal{O}_Y = \mathcal{O}_Z$ and $Z$ is Gorenstein.