Jaffard families and localizations of star operations

We generalize the concept of localization of a star operation to flat overrings; subsequently, we investigate the possibility of representing the set $\mathrm{Star}(R)$ of star operations on $R$ as the product of $\mathrm{Star}(T)$, as $T$ ranges in a family of overrings of $R$ with special properties. We then apply this method to study the set of star operations on a Pr\"ufer domain $R$, in particular the set of stable star operations and the star-class groups of $R$.