We report optical observations of the dissolution behaviour of glycerol/water, soybean oil/hexane, and isobutyric acid (IBA)/water binary mixtures within a glass micromodel built as a 2D regular network of capillary tubes with diameter of 0.2 mm. The micromodel is initially filled with solute and then is horizontally immersed into a thermostatic solvent-filled bath. The micromodel is open at its corners for solute dissolution to occur with no pressure gradients being applied. Our study shows that the solvent penetration into the micromodel is diffusion-dominated in completely miscible binary mixtures (glycerol/water and soybean oil/hexane). This is, however, non-Fickian diffusion with the dissolution rate, dV/dt, being proportional to D1/3t-0.4 for almost the entire duration of the experiment (V is the volume occupied by the solvent, D is the diffusion coefficient, and t is time). For the partially miscible IBA/water mixture the experiments performed at undercritical temperatures revealed that the diffusive transport was negligible despite the mixture being out of its thermodynamic equilibrium. The water phase penetrated into some of the channels, but IBA was never completely displaced/dissolved from the micromodel and numerous interfaces remained visible after very long-time periods.