Suppose that $f:S\rightarrow C$ is a projective morphism between a smooth projective surface to a smooth projective curve over an algebraically closed field. Suppose furthermore that $K_S$ is ample and $f_\ast \mathcal{O}_S=\mathcal{O}_C$. Denote a general fiber by $F$ and let $m>>0$: is there a formula to express the asymptotic behaviour of the $m^{th}$-plurigenus $h^0(S,\mathcal{O}_S(mK_S))$ in terms of (among other things) the genera $g(F)$ and $g(C)$?
Edit: For example, suppose $S=C\times F$ where $F,C$ are smooth curves of genus $\geq 2$. Then, by Riemann-Roch for surfaces
$$h^0(S,\mathcal{O}_S(mK_S))=\chi(\mathcal{O}_S)+4m(m-1)(g_F-1)(g_C-1)$$ provided $m>>0$. I am asking whether an analogous formula holds for $S$ as in the question (or if there are conditions under which something like this holds).