Finiteness of function field-valued points on exceptional Shimura varieties

Let $C\mathrm{/}k$ be a smooth curve over a finite field of characteristic $p>0$. We prove that there are finitely many principally polarized abelian schemes of given dimension $g$ over $C$ up to $p$-power isogeny. For curves over $\bar{k}$, we prove that the moduli space of such abelian schemes is finite type up to $p$-power isogeny. Moreover, we generalize this result to arbitrary (not necessarily abelian type) Shimura varieties $S$ and sufficiently large primes $p$ in terms of $S$: The space of generically ordinary morphisms $C\to S_k$ (resp. $C\to S_{\bar{k}})$ is finite (resp. finite type) up to $p$-Hecke orbits.