I am curious about the analogue of this question, as stated in the title. Namely,

If $Z \subset X$ is a closed subscheme and $Y \to Z$ is faithfully flat (let's also say of finite presentation), can we find a map $Y' \to X$ which is flat, whose image contains $Z$ (and is thus a neighborhood of $Z$), and whose restriction to $Z$ is the given map?

This may be too strong; i.e. just as for étale maps, in the other question, this may be true only Zariski locally on $X$. That's fine too. In that case, in its most basic form it becomes a question of commutative algebra:

If $R$ is a ring, $I \subset R$ an ideal, and $S$ a faithfully flat $R/I$-algebra, is there a set of elements $f \in R$ such that $\sum (R/I) f = R/I$ and a corresponding set of faithfully flat $R_f$-algebras ${}_f\widetilde{S}$ such that ${}_f \widetilde{S}/I_f \cong S_f$ for each $f$?

Also, it would be nice to know that, in the event that this is true, it preserves finiteness hypotheses like "finitely presented".

For those who are curious about my motivations: I want to show that "$!$ pushforwards (resp. pullbacks) commute with $*$ pushforwards (resp. pullbacks)" under appropriate circumstances, those being when the $!$'s are along closed immersions and the $*$'s are along faithfully flat covers or vice-versa. At the very least I need to be able to switch the order in which maps of these types appear; that is, rewrite a cover of a closed immersion as a closed immersion into a cover, as my question asks.