  We investigate the spin transport properties of a topological magnon
insulator, a magnetic insulator characterized by topologically nontrivial bulk
magnon bands and protected magnon edge modes located in the bulk band gaps.
Employing the Landau-Lifshitz-Gilbert phenomenology, we calculate the spin
current driven through a normal metal$|$topological magnon insulator$|$normal
metal heterostructure by a spin accumulation imbalance between the metals, with
and without random lattice defects. We show that bulk and edge transport are
characterized by different length scales. This results in a characteristic
system size where the magnon transport crosses over from being bulk-dominated
for small systems to edge-dominated for larger systems. These findings are
generic and relevant for topological transport in systems of nonconserved
bosons.
