Based on measurements over a period of more than 650 days, we investigated the long-term stability of superconducting Mo-Cu bilayer films in the laboratory environment. The samples are based on sputtered films, which were grown for fabricating transition-edge sensor (TES) arrays. The results show an increasing trend of their superconducting transition temperature (Tc) and transition width with time. There appears to be a turning point between 150 to 200 days, after which the rate of increasing becomes smaller. We suspect that oxidation occurred at the interface of the bilayers, through exposed edges, until the edges are fully oxidized. If proven, the slower rising trend would reflect the subsequent evolution of the bilayer films through, e.g., further oxidation of the interface, which might change Tc through the proximity effect. We quantified the proximity effect with our bilayer samples of different Cu- to-Mo thickness ratios. As a function of the thickness ratio, the measured Tc values are fitted well with the Martinis equation, provided that the transmission coefficent (between the layers) and Tc,Mo of the Mo layer are both allowed to vary. Based the best-fit model, we found that a change of about 1% in the transmission coefficient would be sufficient to account for the slower variation in the bilayer transition temperature. We also noticed that the best-fit value of Tc,Mo is significantly higher than that we measured of our bare Mo films. This could also be explained by surface oxidation of the Mo film samples. The implications of our results on TES-related applications are discussed.