An experimental and theoretical study is presented of coherent effects in electron transport in the doublebarrier SINIS junctions ͑where S, I, and N denote a superconductor, insulator, and normal metal, respectively͒. The appearance of a steplike subgap structure in the current-voltage characteristics of the Nb/Al/AlO x /Al/AlO x /Al/Nb superconducting junctions at a voltage Vϳ⌬ Nb /e ͑where ⌬ Nb is the superconducting energy gap of Nb͒ is interpreted as a manifestation of a nonequilibrium supercurrent at finite dc bias voltage ͑Finite-Bias Josephson Effect͒. The origin of this effect lies in the energy-band structure associated with a set of macroscopic quantum states characteristic of a SINIS junction. Specifically, the junction can have an energy level near energy ⌬ Nb , which provides an additional channel for dc Josephson current at V ϳ⌬ Nb /e. In addition, sharp features in the conductivity at a voltage near the gap-sum voltage were observed in both SINIS and SINININIS junctions, implying correlated quasiparticle tunneling in multiple-barrier junctions. Our theoretical model provides a good qualitative description of the quasiparticle conductivity, including narrow peaks at finite voltage and a zero-voltage anomaly observed on some samples, and suggests an alternative explanation of a feature interpreted earlier as gap-difference feature associated with the tunneling extraction of quasiparticles from the middle Al layer.