  The transport properties of strongly interacting fermionic systems can reveal
exotic states of matter, but experiments and theory have predominantly focused
on bulk systems in the hydrodynamic limit describable with linear response
coefficients such as electrical and thermal conductivity. In a ballistic
channel connecting two superfluid reservoirs, recent experiments revealed a
far-from-equilibrium regime beyond linear hydrodynamics where particle and
entropy currents respond nonlinearly to biases of chemical potential and
temperature, and their ratio is robust to the channel geometry. However, the
origin of this robustness and its relation to the strong interparticle
interactions remain unknown. Here, we study the coupled transport of particles
and entropy tuning the interaction across the Bardeen-Cooper-Schrieffer to
Bose-Einstein condensate (BCS-BEC) crossover, the reservoir degeneracy across
the superfluid phase transition, as well as the local potentials and
confinement of the channel. Surprisingly, the entropy advectively transported
per particle depends only on the interactions and reservoir degeneracy and not
on the details of the channel, suggesting that this property has its origin in
the universal equilibrium properties of the reservoirs. In contrast, the
magnitudes of the advective and diffusive entropy currents vary significantly
with the channel details. The advective current increases monotonically towards
the BEC side, which can be largely explained by the estimated superfluid gap in
the channel. The Wiedemann-Franz law that links the advective and diffusive
currents in Fermi liquids is most egregiously violated at unitarity, suggesting
a change in the nature of the excitations responsible for entropy diffusion
near unitarity. These observations pose fundamental questions regarding
transport phenomena in strongly interacting Fermi systems far from equilibrium.
