Let $k$ be a field and let $X$ be a hyperbolic curve over $k$.
Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$.
I know this statement holds over $k=\mathbf{C}$. In particular, it holds over $k=\overline{\mathbf{Q}}$.
Does it hold over any field $k$?
What about a number field?