  We consider a lcsc group G acting properly on a Borel space S and measurably
on an underlying sigma-finite measure space. Our first main result is a
transport formula connecting the Palm pairs of jointly stationary random
measures on S. A key (and new) technical result is a measurable disintegration
of the Haar measure on G along the orbits. The second main result is an
intrinsic characterization of the Palm pairs of a G-invariant random measure.
We then proceed with deriving a general version of the mass-transport principle
for possibly non-transitive and non-unimodular group operations first in a
deterministic and then in its full probabilistic form.
