Let $\pi:\mathfrak{Y}\to B$ be a projective and flat morphism, with $B$ a smooth projective curve over $k$. If $\mathfrak{L}\in\Pic(\mathfrak{Y})$ is ample on the generic fiber of $\pi$, then there exists an ample line bundle $H$ on $B$ such that $\mathcal{O}_{B}(mH)\otimes\pi_{*}\mathcal{O}_{\mathfrak{X}}(m\mathfrak{L})$ is globally generated for all $m$ sufficiently large and divisible.