For a variety $X$ over a field $k$, we define $Br(X) = H^2_{et}(X,\mathbb{G}_m)$.
Suppose $X$ is a smooth variety (finite type, separated) over an algebraically closed field $k$ together with a morphism $X \to \mathbb{A}^1_k$. Let $X_{\eta}$ be the generic fiber of this morphism, which is a smooth variety over $k(t)$.
Is it true that there is always an injection $Br(X) \to Br(X_\eta)$? How is it constructed, why do you need smoothness to get injectivity (if this is true, of course)?