We present a comprehensive study on the nonequilibrium properties of superconducting nanoconstriction junctions in steady-state and time-dependent dynamic regimes. By measuring the current-voltage characteristics of single constrictions and micro-SQUID devices with nanobridges, we observe a series of distinct voltage-step jumps in the dissipative state, which are associated with the appearance of the excess current. Through detailed analysis, we identify different mechanisms that contribute to the enhancement of superconductivity under varying bias voltages. In the time-dependent dynamic regime where the bias voltage V significantly exceeds the superconducting gap voltage (V ≫ Δ/e), the nanoconstriction behaves as a single phase-slip center (PSC). The voltage steps observed in this regime signify the phase-slipping dynamics at PSC, which effectively mitigates the self-heating effects. Conversely, in the steady-state regime where V is lower than the gap voltage, the voltage steps arise from multiple Andreev reflections. We demonstrate that the interplay of Andreev quasiparticles and Josephson supercurrent plays a crucial role in restoring the phase coherence of the dissipation currents. Our findings provide an insightful understanding of nonequilibrium quasiparticle relaxation dynamics in superconducting nanometallic weak links and offer practical implications for potential applications.