We prove that the Aizenman–Contucci relations, well known for fully connected spin glasses, hold in diluted spin glasses as well. We also prove more general constraints in the same spirit for multi-overlaps, systematically confirming and expanding previous results. The strategy that we employ makes no use of self-averaging, and allows us to generate hierarchically all such relations within the framework of random multi-overlap structures. The basic idea is to study, for these structures, the consequences of the closely related concepts of stochastic stability, quasi-stationarity under random shifts, factorization of the trial free energy. The very simple technique allows us to prove also the phase transition for the overlap: it remains strictly positive (on average) below the critical temperature if a suitable external field is first applied and then removed in the thermodynamic limit. We also deduce, from a cavity approach, the general form of the constraints on the distribution of multi-overlaps found within quasi-stationary random multi-overlap structures.