Properness is local on the target

Suppose $f$ is proper and fix $i$. By [F1], $f$ is separated, so [F5] makes ${\mathrm{\Delta }}_{X\mathrm{/}S}$ a closed immersion. The opens $(X{\times }_{S}X){\times }_S{S}_i$ cover $X{\times }_{S}X$; [F6] makes each restriction a closed immersion, and [F7] identifies it with ${\mathrm{\Delta }}_{X_i\mathrm{/}{S}_i}$. Thus $f_i$ is separated.

Properness of $f$ gives finite type, hence local finite type and quasi-compactness by [F2]. For a point of $X_i$, refine a local finite-type affine chart on the source and target to affine opens contained in $X_i$ and $S_i$: first take a principal target neighbourhood inside the target chart and $S_i$, then a principal source neighbourhood inside its inverse image. The localized ring map remains of finite type by [F9] and the affine-open basis [F10, F11, F12]. The base-change assertion in [F3] makes $f_i$ quasi-compact, so [F2] makes it finite type.

For any $T\to S_i$, regard $T$ as an $S$-scheme by composition. By [F8], the base change of $f_i$ to $T$ is canonically the base change of $f$ to $T$. Universal closedness of $f$ therefore makes the base change of $f_i$ closed, so $f_i$ is universally closed.

Now suppose every $f_i$ is proper. By [F1] each is finite type, so the local finite-type charts over the cover $X={\bigcup }_i{X}_i$ show that $f$ is locally of finite type. For quasi-compactness, take all affine open subschemes $V\subseteq S_i$ for all $i$. They cover $S$: given $s\in S_i$, an affine neighbourhood $U=\mathrm{Spec}A$ exists by [F10]; the open $U\cap S_i$ contains $s$, so [F11] gives a principal open $D(a)$ with $s\in D(a)\subseteq U\cap S_i$, and [F12] makes it affine. For each such $V\subseteq S_i$, $f^{-1}(V)\to V$ is a base change of the quasi-compact map $f_i$, hence quasi-compact by [F3]. The affine-cover criterion in [F3] shows that $f$ is quasi-compact, so [F2] makes it finite type. The cover consists of all qualifying affine opens; no simultaneous choices are made.

By [F1], each $f_i$ is separated, so each ${\mathrm{\Delta }}_{X_i\mathrm{/}{S}_i}$ is a closed immersion by [F5]. The opens $(X{\times }_{S}X){\times }_S{S}_i$ cover $X{\times }_{S}X$, and [F7] identifies the restriction of ${\mathrm{\Delta }}_{X\mathrm{/}S}$ to each with ${\mathrm{\Delta }}_{X_i\mathrm{/}{S}_i}$. By [F6], ${\mathrm{\Delta }}_{X\mathrm{/}S}$ is a closed immersion, so $f$ is separated.

Fix any $T\to S$ and closed $C\subseteq X{\times }_{S}T$. The opens $T_i=T{\times }_S{S}_i$ cover $T$. By [F8], the restriction of $X{\times }_{S}T$ over $T_i$ is canonically $X_i{\times }_{S_i}{T}_i$. By [F1], $f_i$ is universally closed, so the image of the restricted closed subset is closed in $T_i$; this image is exactly $f_T(C)\cap T_i$. A subset whose intersections with an open cover are closed is closed. Thus $X{\times }_{S}T\to T$ is closed. Since $T$ and $C$ were arbitrary, $f$ is universally closed by [F4].

Steps 1.4, 1.5, and 1.6 give finite type, separatedness, and universal closedness for $f$, so [F1] makes $f$ proper. Steps 1.1–1.3 prove the other direction. If $X=\mathrm{\varnothing }$, every restriction is empty and proper; if $S=\mathrm{\varnothing }$, then also $X=\mathrm{\varnothing }$ and the cover may be empty. For a one-member cover $\{S\}$ the restriction is $f$ itself. Empty cover members and affine charts with coordinate ring $0$ add no points.