We investigate the transport properties of a superfluid Fermi gas confined in a ring trap with a moving potential barrier across the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover. Employing time-dependent Bogoliubov–de Gennes (BdG) equations, we simulate the dynamics of a Josephson junction biased by both DC and AC currents. Over a wide range of interaction strengths, the barrier-velocity dependence of the chemical potential difference exhibits low-order plateau structures, consistent with Shapiro steps, with fitted levels close to integer multiples of $\hslash \omega /2$ within the phase-coherent regime. This factor of $1/2$ reflects our convention of defining the chemical potential per single fermion in the BdG framework. Microscopic analysis reveals that these fundamental steps originate from synchronized phase slips mediated by periodic soliton generation at the barrier. Our findings clarify the role of interaction regimes in the nonequilibrium phase dynamics of ring-trapped fermionic superfluids and provide microscopic insights relevant to future studies of atomtronic systems with nontrivial topology.