Let $X,Y$ be defined over the field $k$ and  take $f$ to be the structure map to ${\rm Spec}\, k$. Then let $E\to G$ be a surjective morphism of sheaves that is not surjective on global sections, e.g., $$\mathcal O_{\mathbb P^1}(-1)\oplus \mathcal O_{\mathbb P^1}(-1)\to \mathcal O_{\mathbb P^1}.$$
Then $f_*$ is just $H^0$ and the desired statement is false.