  We consider heterostructures consisting of regions with a continuous symmetry
in contact with regions wherein the symmetry is spontaneously broken. The
low-frequency dynamics of the corresponding order parameter are shown to induce
nonequilibrium transport, a ``pumping,'' out of the symmetry-broken regions,
which is governed by the generator of the broken-symmetry operator. This
pumping damps Goldstone-mode excitations and transfers them beyond traditional
(static) proximity length scales. Our general conclusions are discussed for
specific examples of order parameters in helimagnets, charge/spin-density
waves, superconductors, and ferromagnets. We carry out a detailed calculation
of such pumping for spiral magnetic orders in helimagnets possessing a duality
in the representation of its symmetry-broken states.
