Let $S$ a qcqs scheme, and let $f : X := \varprojlim\_j X\_j \to S$ be an inverse limit of qcqs schemes $f\_j : X\_j \to S$ with affine transition maps.


Suppose $f$ is (P). Is $f\_j$ also (P) for all $j$ large enough?


(1) P = flat;


(2) P = finite;


(3) P = quasi-finite.



Relative approximation of morphisms
No (in general).
Let $A$ be a non-zero ring and let $S = \mathrm{Spec}(A[T]/(T^2))$. Let $M$ be a free $A$-module of infinite rank, viewed as an $A[T]/(T^2)$-module via the section $A[T]/(T^2) \rightarrow A$ which sends $T$ to $0$. 


Take $X\_j = \mathrm{Spec}(A[T]/(T^2) \oplus M)$, where $m^2 = 0$ for any $m \in M$, with affine transition maps given by the composition
$$
A[T]/(T^2) \oplus M \rightarrow A[T]/(T^2) \rightarrow A[T]/(T^2) \oplus M.
$$
The limit $X \rightarrow S$ is an isomorphism, hence satisfies $(1),(2),(3)$, while each $X\_j$ satisfies none of the properties $(1),(2),(3)$.


