Dear Giovanni, here is a point of view orthogonal to the examples you mention     (although I am not sure they will satisfy your wish that they  be "stranger"... ). 

Given a morphism $X \to S$, instead of **enlarging** $S$ to some bigger scheme $S'$,  take instead a subscheme $S'\subset S$ : base change will then correspond to scheme-theoretic inverse image.
In the affine case, $SpecB \to SpecA$, base change will lead to $Spec B/IB \to Spec A/I$ : extension corresponds to quotient.

In particular for a closed point $P \in S$ you will get the fibre over $P$. And if the base scheme is integral the fibre over the generic point $\eta$ of $S$ will give you the *generic* fibre of the morphism, which is a scheme over the rational function field $Rat(S)= k(\eta)$ of $S$.

Probably you know all this but I thought it might be useful to have a reminder that contrary to our unconscious bias due to the terminology , base change  and ring extensions may lead to **smaller** objects : subschemes and quotient rings.