It is a nice exercise in combinatorics to show that every 2d-regular finite graph arises as a Schreier graph of the free group Fd. I will present generalizations of this fact to a measurable setting, as well as some examples showing the limitations. I will formulate these results using both the language of unimodular random networks and that of (p.m.p.) graphings, which are two sides of the same coin. Partially joint work with Ferenc Bencs and Aranka Hrušková.