The evolution of small disturbances to a horizontal interface separating two miscible liquids is examined. The aim is to investigate how the interfacial mass transfer affects development of the Rayleigh-Taylor instability and propagation and damping of the gravity-capillary waves. The phase-field approach is employed to model the evolution of a miscible multiphase system. Within this approach, the interface is represented as a transitional layer of small but nonzero thickness. The thermodynamics is defined by the Landau free energy function. Initially, the liquid-liquid binary system is assumed to be out of its thermodynamic equilibrium, and hence, the system undergoes a slow transition to its thermodynamic equilibrium. The linear stability of such a slowly diffusing interface with respect to normal hydro- and thermodynamic perturbations is numerically studied. As a result, we show that the eigenvalue spectra for a sharp immiscible interface can be successfully reproduced for long-wave disturbances, with wavelengths exceeding the interface thickness. We also find that thin interfaces are thermodynamically stable, while thicker interfaces, with the thicknesses exceeding an equilibrium value, are thermodynamically unstable. The thermodynamic instability can make the configuration with a heavier liquid lying underneath unstable. We also find that the interfacial mass transfer introduces additional dissipation, reducing the growth rate of the Rayleigh-Taylor instability and increasing the dissipation of the gravity waves. Moreover, mutual action of diffusive and viscous effects completely suppresses development of the modes with shorter wavelengths.