Let $f: X \to Y$ be a finite morphism of varieties with $Y$ smooth. Is there a projection formula for $f$ and $H^{i}_{et}(-,\mathbf{Z}(1)) = H^i(-,\mathbf{G}_m)$?

Background: I want to show that for a relative curve $\pi: C \to X$ ($C$, $X$, $\pi$ smooth) with a quasi-section $Y \to C$ closed immersion with $Y \to X$ finite of degree $d$, the kernel of $\pi^*: H^i(X,\mathbf{G}_m) \to H^i(C,\mathbf{G}_m)$ is annihilated by $d$.