So for certain base changes, it's clear that 'base change' really means base change from high school : for example, if a curve is defined over a field, it will be of course defined over any extension of that field, and 'base change' as defined for schemes agrees with that, essentially because of properties of tensor product. Similarly, if we define projective space over $\mathbb{Z}$, or the general linear group scheme over $\mathbb{Z}$, then changing base by the unique $Spec(A)\to Spec(Z)$ (for any ring $A$) gives projective space over $A$ and the general linear group scheme over $A$ as well.

It would be nice to see some examples of base change of $X/S$ through some stranger change of base maps $S'\to S$, and to see how $X/S$ is deformed by $S'\to S$ in ways that are 'fundamentally' not like the above ; i.e., the examples I listed above don't really capture the 'spirit' of base change in general. I'm looking for a list of other central geometric ideas behind the notion of base change that go beyond the above ideas.

For example, if $X/S$ is finite etale, $s$ in the image, then changing base by the strict localization at $s$ gives a trivial finite étale covering (i.e. the base change is a finite union of copies of the strict localization at $s$), thus showing that (surjective?), finite étale maps are locally trivial for the étale topology (and therefore the name 'étale covering' is meaningful in this sense).

Can someone give a motivated discussion of the geometry of base change ? E.g., examples-based observation and pondering.