Deliquescence is the very first stage of heterogeneous unary condensation on solid soluble nuclei; at this stage, a solid nucleus dissolves in a liquid film formed by the molecules of the vapor condensing on the nucleus. As recently shown [Djikaev , J. Phys. Chem. B 105, 7708 (2001)], deliquescence in the atmosphere may occur in a fluctuational mode at relative humidities lower than the "deliquescence point" (at this point deliquescence becomes barrierless). Developing the kinetics of fluctuational deliquescence, we derive a two-dimensional kinetic equation, governing the time evolution of the droplet distribution. When solid soluble nuclei consist of molecules of a single species, in the vicinity of the saddle point of the free energy surface of deliquescence this equation has the form of the kinetic equation of binary nucleation. A quasisteady-state solution for this equation is obtained by using the method of complete separation of variables. An expression for the average deviation of the stable variable of state of a droplet from its equilibrium value is also derived. It shows that the film of liquid solution, forming around the nucleus during its deliquescence, is not in equilibrium neither with the vapor nor with the solid core. A recipe is proposed to construct the time evolution of the quasisteady state caused by the depletion of undeliquesced nuclei in the system. The theoretical results are illustrated by numerical calculation for the deliquescence of model particles in a water vapor.