Bicontinuous particle-stabilized emulsions (bijels) are unique soft materials that combine the bulk properties of two immiscible fluids into a single interconnected structure. This structure is achieved through the formation of two interwoven fluid networks, stabilized by an interfacial layer of colloidal particles. Bijels with submicron-scale domain networks can be synthesized via solvent transfer-induced phase separation (STrIPS). However, the fluid network structure in STrIPS-bijels tends to degrade over time, limiting their practical applications. In this study, we identify that the destabilization of STrIPS-bijels is driven by the exchange of matter between the bijel and the surrounding oil phase. Confocal laser scanning microscopy reveals that over time, aqueous components from the bijel dissolve into the surrounding oil, leading to an inward flow of oil into the bijel. This process disrupts the fluid bicontinuous structure within hours. To extend the stability of the bijel to several weeks, we explore strategies to reduce the dissolution of the aqueous phase and the inflow of oil. Specifically, we investigate the effects of the oil's chemical composition and properties, as well as modifications to the surface chemistry of the supporting glass substrates. Our results show that while the particle scaffold of STrIPS-bijels exhibits long-term stability, the maintenance of the fluid bicontinuous network depends on minimizing the loss of aqueous components. The enhanced control over the stability of the fluid bicontinuous structure developed in this work is critical for advancing STrIPS-bijels as functional materials for applications in catalysis, separations, and energy storage.