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 et al. [Phys. Rev. Lett. 119, 056804 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.056804], where a stochastic approach is justified. We obtain the temperature and thickness dependence of the structural spin Seebeck coefficient S and magnon conductance G. In their evaluation, we incorporate effects from field- and temperature-dependent spin conserving intermagnon scattering processes. Furthermore, the interfacial spin transport is studied by evaluating the contact magnon conductances in a microscopic model that accounts for the sublattice symmetry breaking at the interface. We find that while intermagnon 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.