  We develop a Green's function formalism for spin transport through
heterostructures that contain metallic leads and insulating ferromagnets. While
this formalism in principle allows for the inclusion of various magnonic
interactions, we focus on Gilbert damping. As an application, we consider
ballistic spin transport by exchange magnons in a metal-insulator-metal
heterostructure with and without disorder. For the former case, we show that
the interplay between disorder and Gilbert damping leads to spin current
fluctuations. For the case without disorder, we obtain the dependence of the
transmitted spin current on the thickness of the ferromagnet. Moreover, we show
that the results of the Green's function formalism agree in the clean and
continuum limit with those obtained from the linearized stochastic
Landau-Lifshitz-Gilbert equation. The developed Green's function formalism is a
natural starting point for numerical studies of magnon transport in
heterostructures that contain normal metals and magnetic insulators.
