In EGA IV, Chapter 8, projective systems of schemes (and morphisms between them) are considered. Let $(S_{\lambda})_{\lambda \in L}$ be a projective system of schemes and let $S$ be the projective limit of this system. My question is this. In Théorème 8.10.5, it is stated that for every property P in a given list of properties (starting with: isomorphism, monomorphism, immersion...), and under some additional conditions that I won't mention right now, if a morphism $f : X \rightarrow Y$ of $S$-schemes with property P can be "spreaded out" to a morphism $f_{\alpha} : X_{\alpha} \rightarrow Y_{\alpha}$ of $S_{\alpha}$-schemes for some $\alpha \in L$, then we can choose $\alpha$ in such a way that $f_{\alpha}$ has property P. The list of properties P given in EGA for which the Théorème works, however, does not contain flatness. Is there a reason for this? I.e., is there a counter-example for the statement of the theorem if we include flatness in the list? If so, are there any reasonable extra assumptions that we can put into the theorem so that it also works for flatness?

I will the describe the specifics of the situation that I am most interested in in more detail. Suppose that $R$ is a commutative ring, then let $R_i$ be the directed system of its finitely generated subrings. Put $S_i := \textrm{Spec}(R_i)$, so that $(S_i)_{i \in I}$ is a projective system of affine schemes. I have a morphism $f : X \rightarrow S$ that is proper, flat and of finite presentation. Does $f$ necessarily arise as the base-change of some $f_i : X_i \rightarrow S_i$ that is also proper, flat and of finite presentation?