We investigate the unsteady phase transition problem with kinetic interface conditions in rarefied gas deposition and sublimation. We couple the R13 equations of macroscopic gas transport with the heat equation to model the interaction between the gas and the solid phase. The kinetic interface conditions are derived from thermodynamic principles and provide an accurate description of the phase transition process. We employ the Finite Element method with a remeshing approach to solve the coupled system and track the moving interface. Our numerical results demonstrate the effectiveness of the proposed method in capturing the complex dynamics of the phase transition process, including the effects of kinetic interface conditions.