Let $X$ be a scheme which is smooth over a noetherian base scheme $S$. Let $(\mathcal F,\nabla)$ be a flat vector bundle, i.e. $\mathcal F$ is a vector bundle of finite rank on $X$ and $\nabla$ is an integrable connection relative $S$.

Is there a notion in the literature of "unipotent" flat vector bundle and morphisms between them? And does this give an abelian category with objects isomorphism classes of such $(\mathcal F, \nabla)$? I haven't seen a clean formal approach to this in the literature.

I don't think it is abelian in the above wide context, but probably for $X$ being a smooth variety over a field or an abelian variety over a field.