  We introduce and study the question how can stable birational types vary in a
smooth proper family. Our starting point is the specialization for stable
birational types of Nicaise and the author and our emphasis is on stable
birational types of hypersurfaces. Building up on the work of Totaro and
Schreieder on stable irrationality of hypersurfaces of high degree, we show
that smooth Fano hypersurfaces of large degree over a field of characteristic
zero are in general not stably birational to each other. In the appendix Claire
Voisin proves a similar result in a different setting using the Chow
decomposition of diagonal and unramified cohomology.
