  Consider the Hamiltonian action of a torus on a transversely symplectic
foliation that is also Riemannian. When the transverse hard Lefschetz property
is satisfied, we establish a foliated version of the Kirwan injectivity
theorem, and use it to study Hamiltonian torus actions on transversely K\"ahler
foliations. Among other things, we prove a foliated analogue of the
Carrell--Liberman theorem. As an application, this confirms a conjecture raised
by Battaglia--Zaffran on the basic Hodge numbers of symplectic toric
quasifolds. Our methods also allow us to present a symplectic approach to the
calculation of the Betti numbers of symplectic toric quasifolds as
diffeological spaces.
