All varieties are over $\mathbb{C}$.

Let $X$ be a variety and $\pi \colon E \to X$ a geometric vector bundle. So $ \pi $ is affine. Then certainly the assignment $ M \mapsto \pi_*M $ defines an equivalence between quasi-coherent $\mathcal{O}_E$-modules and quasi-coherent $\pi_*\mathcal{O}_E$-modules.

Now $\mathbb{C}^{\times}$ acts on $E$ via dilation of the fibres of $\pi$. So $\pi_* \mathcal{O}_E$ acquires a grading. Is it true that $M \mapsto \pi_* M$ gives an equivalence between $\mathbb{C}^{\times}$-equivariant quasi-coherent $\mathcal{O}_E$-modules and graded quasi-coherent $\pi_*\mathcal{O}_E$-modules?

If this is true, does it generalize to replacing $\pi$ being a vector bundle with $E$ just equipped with a $\mathbb{C}^{\times}$-action, $\pi$ affine and $\mathbb{C}^{\times}$-equivariant, $\mathbb{C}^{\times}$ acting on $X$ trivially?