Geometric local systems on very general curves and isomonodromy

We show that the minimum rank of a non-isotrivial local system of geometric origin, on a suitably general $n$-pointed curve of genus $g$, is at least $2\sqrt{g+1}$. We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformation, which additionally answers questions of Biswas, Heu, and Hurtubise.