Are there any interesting cases (interesting here is interpreted rather loosely here) where you can show $X$ has property $P$ whenever all $X(K)$ have property $P$ where $K$ runs through all fields?

 

Motivation:
Sometimes the field valued points of a scheme are much easier to understand than general $R$ points. For example a split reductive group $G$ you have the Bruhat decomposition of $G(k)$ which tell you the double cosets for a Borel $B \subset G$ are indexed by permutation matrices. But in general the cosets in $G(R)$ for a $k$-alebra $R$ are more complicated.