An approximate mathematical model of contact melting of an unfixed material in elastic cylindrical and spherical capsules is developed. Since the density of the solid is higher than that of the melt, the melting solid resides at the bottom supported by a thin layer of the generated, convecting, melt, and the capsule swells. The main characteristic scales and non-dimensional parameters, which describe the principal features of the melting process and the liquid flow, are found. Linearisation with regard to the Stefan number as well as the small difference between the densities of the solid and liquid enables us to derive a closed-form evolution equation for the motion of the solid, which also determines the melting rate, Numerical solution of the evolution equation shows that the swelling of the capsule during melting, which is caused by the decrease of the density during phase transition, leads to slowing down of the melting process. This effect is due to flattening of the lower surface of the capsule, which entails fall of the pressure along with thickening of the molten layer. The latter determines the decrease of the melting rate.