  The Ratner property, a quantitative form of divergence of nearby
trajectories, is a central feature in the study of parabolic homogeneous flows.
Discovered by Marina Ratner and used in her 1980th seminal works on horocycle
flows, it pushed forward the disjointness theory of such systems. In this
paper, exploiting a recent variation of the Ratner property, we prove new
disjointness phenomena for smooth parabolic flows beyond the homogeneous world.
In particular, we establish a general disjointness criterion based on the
switchable Ratner property. We then apply this new criterion to study
disjointness properties of smooth time changes of horocycle flows and smooth
Arnol'd flows on the torus, focusing in particular on disjointness of distinct
flow rescalings. As a consequence, we answer a question by Marina Ratner on the
Moebius orthogonality of time-changes of horocycle flows. In fact, we prove
Moebius orthogonality for all smooth time-changes of horocycle flows and
uniquely ergodic realizations of Arnol'd flows considered.
