This is vaguely similar to, but quite different from, this question.
In the above linked question the focus is on fields of large cardinality per se. Here we fix a base field $k$ (say algebraically closed and of characteristic zero), and we consider "objects" $\xi$ over a base $k$-variety or scheme $S$ (for example, $\xi$ could be a morphism $X\to S$, a family of curves, a coherent sheaf on $S$, a flat family of coherent sheaves on some $X$ parametrized by $S$, etc).
This is potentially a "big list" sort of question (if it's not even too broad to be legit).
I would like to ask about interesting cases in which facts and phenomena about $\xi$ can be deduced from more classical algebraic geometry about the base change $\xi_\eta$ of $\xi$ to a/the generic point $\eta$ of $S$. So $\xi_\eta$ will be an "object" over a field $K=\kappa(\eta)$ larger than the base field.
For example, it could be a nontrivial fact about elliptic surfaces over $k$ that are deduced from the geometry of elliptic curves over $K$
Also: are there cases in which it is "natural" to consider (objects over) the algebraic closure of $K$?