  Given a Hamiltonian action of a proper symplectic groupoid (for instance, a
Hamiltonian action of a compact Lie group), we show that the transverse
momentum map admits a natural constant rank stratification. To this end, we
construct a refinement of the canonical stratification associated to the Lie
groupoid action (the orbit type stratification, in the case of a Hamiltonian
Lie group action) that seems not to have appeared before, even in the
literature on Hamiltonian Lie group actions. This refinement turns out to be
compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson
stratification of the orbit space, each stratum of which is a regular Poisson
manifold that admits a natural proper symplectic groupoid integrating it. The
main tools in our proofs (which we believe could be of independent interest)
are a version of the Marle-Guillemin-Sternberg normal form theorem for
Hamiltonian actions of proper symplectic groupoids and a notion of equivalence
between Hamiltonian actions of symplectic groupoids, closely related to Morita
equivalence between symplectic groupoids.
