Incomparable actions of free groups

Suppose that X is a Polish space, E is a countable Borel equivalence relation on X, and mu is an E-invariant Borel probability measure on X. We consider the circumstances under which for every countable non-abelian free group Gamma, there is a Borel sequence. (.(r))(r is an element of R) of free actions of Gamma on X, generating subequivalence relations E-r of E with respect to which mu is ergodic, with the further property that (E-r)(r is an element of R) is an increasing sequence of relations which are pairwise incomparable under mu-reducibility. In particular, we show that if E satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on X, generating a subequivalence relation of E with respect to which mu is ergodic.