  We theoretically investigate laser-induced spin transport in metallic
magnetic heterostructures using an effective spin transport description that
treats itinerant electrons and thermal magnons on an equal footing.
Electron-magnon scattering is included and taken as the driving force for
ultrafast demagnetization. We assume that in the low-fluence limit the magnon
system remains in a quasi-equilibrium, allowing a transient nonzero magnon
chemical potential. In combination with the diffusive transport equations for
the itinerant electrons, the description is used to chart the full spin
dynamics within the heterostructure. In agreement with recent experiments, we
find that in case the spin-current-receiving material includes an efficient
spin dissipation channel, the interfacial spin current becomes directly
proportional to the temporal derivative of the magnetization. Based on an
analytical calculation, we discuss that other relations between the spin
current and magnetization may arise in case the spin-current-receiving material
displays inefficient spin-flip scattering. Finally, we discuss the role of
(interfacial) magnon transport and show that, a priori, it cannot be neglected.
However, its significance strongly depends on the system parameters.
