Let $\mathcal C$ be the class of morphisms $f\colon U\to V$ of schemes such that for every proper map $g\colon X\to Y$ between schemes and every commutative solid square

there exists a lift $h$ making the diagram commute.

Is there a nice description of elements of $\mathcal C$? Or at least of some subclass of $\mathcal C$ beyond the morphisms used in the valuative criterion, where $V$ is the spectrum of a valuation ring and $U$ is the spectrum of its fraction field? I am particularly interested in the case where $V=\mathbb A^1\times S$ and $U=(\mathbb A^1\setminus 0)\times S$, for any scheme $S$. For $S$ the spectrum of a field, this follows from 0BX7 in the Stacks Project, but what about general $S$?