  Spin transport of magnonic excitations in uniaxial insulating
antiferromagnets (AFs) is investigated. In linear response to spin biasing and
a temperature gradient, the spin transport properties of
normal-metal--insulating antiferromagnet--normal-metal heterostructures are
calculated. We focus on the thick-film regime, where the AF is thicker than the
magnon equilibration length. This regime allows the use of a drift-diffusion
approach, which is opposed to the thin-film limit considered by Bender {\it et
al.} 2017, where a stochastic approach is justified. We obtain the temperature-
and thickness-dependence of the structural spin Seebeck coefficient
$\mathcal{S}$ and magnon conductance $\mathcal{G}$. In their evaluation we
incorporate effects from field- and temperature-dependent spin conserving
inter-magnon scattering processes. Furthermore, the interfacial spin transport
is studied by evaluating the contact magnon conductances in a microscopic model
that accounts for the sub-lattice symmetry breaking at the interface. We find
that while inter-magnon scattering does slightly suppress the spin Seebeck
effect, transport is generally unaffected, with the relevant spin decay length
being determined by non-magnon-conserving processes such as Gilbert damping. In
addition, we find that while the structural spin conductance may be enhanced
near the spin flip transition, it does not diverge due to spin impedance at the
normal metal|magnet interfaces.
