Let $f : X := \varprojlim_j X_j \to \text{Spec}(V)$ be an inverse limit of qcqs schemes $f_j : X_j \to\text{Spec}(V)$ with affine transition maps, and assume $V$ is a dvr with maximal ideal $\mathfrak{m}$.

Suppose the closed fiber of $f$ is (P). Is the closed fiber $f_j$ also (P) for all $j$ large enough?

(1) P = flat;

(2) P = finite;

(3) P = quasi-finite.

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.