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, 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.