Quasiparticle and Josephson tunneling measurements were carried out on Nb-based junctions with a thick “transition region” from the superconductor to the oxide, fabricated by using a thermal oxidation procedure. In these junctions, the current-voltage characteristics and temperature variations of the Josephson current have the same general features as in a proximity-effect junction with a thick weak-superconducting layer. The experimental results are analyzed in terms of a proximity-effect model in which the order parameters on both sides of a sandwich are determined self-consistently. Good agreement is attained between theory and experiment, but only by dealing with temperature variations of the energy gap and Josephson current independently. A similar calculation is shown to explain very well the experiments by Broom on Nb/Nb junctions.