After discussing the motivation behind the talk and some necessary preliminaries, we will consider the ``stabilization" of a countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group. Using a new local characterization of the Schmidt property, we show that this always gives rise to a so-called stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. We will also discuss some new structural results for product equivalence relations, which we will obtain using von Neumann algebraic techniques.