  Given a surjective morphism $\pi\colon X\to Y$ of normal varieties satisfying
some regularity hypotheses we prove how to recover a Cox ring of the generic
fiber of $\pi$ from the Cox ring of $X$. As a corollary we show that in some
cases it is also possible to recover the Cox ring of a very general fiber, and
finally we give an application in the case of the blowing-up of a toric fiber
space.
