  Standard quantum master equation techniques such as the Redfield or Lindblad
equations are perturbative to second order in the microscopic system-reservoir
coupling parameter $\lambda$. As a result, characteristics of dissipative
systems, which are beyond second order in $\lambda$, are not captured by such
tools. Moreover, if the leading order in the studied effect is
higher-than-quadratic in $\lambda$, a second-order description fundamentally
fails, even at weak coupling. Here, using the reaction coordinate (RC) quantum
master equation framework, we are able to investigate and classify
higher-than-second order transport mechanisms. This technique, which relies on
the redefinition of the system-environment boundary, allows for the effects of
system-bath coupling to be included to high orders. We study steady-state heat
current beyond second-order in two models: The generalized spin-boson model
with non-commuting system-bath operators and a three-level ladder system. In
the latter model heat enters in one transition and it is extracted from a
different one. Crucially, we identify two transport pathways: (i) System's
current, where heat conduction is mediated by transitions in the system, with
the heat current scaling as $j_q \propto \lambda^2$ to lowest order in
$\lambda$. (ii) Inter-bath current, with the thermal baths directly exchanging
energy between them, facilitated by the bridging quantum system. To the lowest
order in $\lambda$, this current scales as $j_q \propto \lambda^4$. These
mechanisms are uncovered and examined using numerical and analytical tools. We
contend that the RC mapping brings, already at the level of the mapped
Hamiltonian, much insights on transport characteristics.
