  We study the emergence of dissipation in an atomic Josephson junction between
weakly-coupled superfluid Fermi gases. We find that vortex-induced phase
slippage is the dominant microscopic source of dissipation across the BEC-BCS
crossover. We explore different dynamical regimes by tuning the bias chemical
potential between the two superfluid reservoirs. For small excitations, we
observe dissipation and phase coherence to coexist, with a resistive current
followed by well-defined Josephson oscillations. We link the junction transport
properties to the phase-slippage mechanism, finding that vortex nucleation is
primarily responsible for the observed trends of conductance and critical
current. For large excitations, we observe the irreversible loss of coherence
between the two superfluids, and transport cannot be described only within an
uncorrelated phase-slip picture. Our findings open new directions for
investigating the interplay between dissipative and superfluid transport in
strongly correlated Fermi systems, and general concepts in out-of-equlibrium
quantum systems.
