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.
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 $(1),(2),(3)$.
