We apply the Nakajima-Zwanzig approach to open quantum systems to study steady-state transport across generic multilevel junctions coupled to bosonic or fermionic reservoirs. The method allows for a unified diagrammatic formulation in Liouville space, with diagrams being classified according to an expansion in the coupling strength between the reservoirs and the junction. Analytical, approximate expressions are provided up to the fourth order for the steady-state boson transport that generalizes to multilevel systems the known results for the low-temperature thermal conductance in the spin-boson model. The formalism is applied to the problem of heat transport in a qubit-resonator junction modeled by the quantum Rabi model. Nontrivial transport features emerge as a result of the interplay between the qubit-oscillator detuning and coupling strength. For quasidegenerate spectra, nonvanishing steady-state coherences cause a suppression of the thermal conductance.