The present work aims at providing a systematic analysis of the current density versus momentum characteristics for a fermionic superfluid throughout the BCS-BEC crossover, even in the fully homogeneous case. At low temperatures, where pairing fluctuations are not strong enough to invalidate a quasiparticle approach, a sharp threshold for the inception of a back-flow current is found, which sets the onset of dissipation and identifies the critical momentum according to Landau. This momentum is seen to smoothly evolve from the BCS to the BEC regimes, whereby a single expression for the single-particle current density that includes pairing fluctuations enables us to incorporate on equal footing two quite distinct dissipative mechanisms, namely, pair breaking and phonon excitations in the two sides of the BCS-BEC crossover, respectively. At finite temperature, where thermal fluctuations broaden the excitation spectrum and make the dissipative (kinetic and thermal) mechanisms intertwined with each other, an alternative criterion due to Bardeen is instead employed to signal the loss of superfluid behavior. In this way, detailed comparison with available experimental data in linear and annular geometries is significantly improved with respect to previous approaches, thereby demonstrating the crucial role played by quantum fluctuations in renormalizing the single-particle excitation spectrum.