One form of de Franchis theorem for algebraic curves is the following: let $X$ be an algebraic curve (defined over $\mathbb{C}$ say) with genus $g > 1$. Then there are only finitely many (isomorphism classes of) curves $Y$ with genus $g' > 1$ such that there is a non-constant map $f : X \rightarrow Y$.

My question is the following: suppose that $X, X^\prime$ are two curves of the same genus $g > 1$ admitting a dominant map to the same curve $Y$ of genus $h > 1$. What can be concluded about $X,X^\prime$? Can there be infinitely many isomorphism classes of such $X$, say?