  We derive the linearly perturbed matching conditions between a Schwarzschild
spacetime region with stationary and axially symmetric perturbations and a FLRW
spacetime with arbitrary perturbations. The matching hypersurface is also
perturbed arbitrarily and, in all cases, the perturbations are decomposed into
scalars using the Hodge operator on the sphere. This allows us to write down
the matching conditions in a compact way. In particular, we find that the
existence of a perturbed (rotating, stationary and vacuum) Schwarzschild cavity
in a perturbed FLRW universe forces the cosmological perturbations to satisfy
constraints that link rotational and gravitational wave perturbations. We also
prove that if the perturbation on the FLRW side vanishes identically, then the
vacuole must be perturbatively static and hence Schwarzschild. By the dual
nature of the problem, the first result translates into links between
rotational and gravitational wave perturbations on a perturbed
Oppenheimer-Snyder model, where the perturbed FLRW dust collapses in a
perturbed Schwarzschild environment which rotates in equilibrium. The second
result implies in particular that no region described by FLRW can be a source
of the Kerr metric.
