Surjectivity and flatness over DVR's (after Moret-Bailly)

We study morphisms of schemes $f:X\to S$ which are locally of finite type. We present conditions under which there exists a morphism $g:S^{\mathrm{\prime }}\to X$ of $S$–schemes such that $f\circ g$ is the canonical morphism $S^{\mathrm{\prime }}\to S$. Furthermore, we exhibit situations in which $f$ is flat surjective. Our results are mostly concerned with $S$ being the spectrum of a DVR.