We considered the shape and crystallographic orientation evolution of the nanoparticles attached to the curved rigid substrate at elevated temperatures at which the particle atoms move by diffusion along the particle surface and along the particle-substrate interface. We developed a general thermodynamic method of calculating the chemical potential of the atoms of nanoparticle located at the particle-substrate interface. This chemical potential can be determined by calculating the change of total surface and interface energy of the system caused by two infinitesimal orthogonal translations and one rigid rotation of the particle. Once the distribution of the chemical potential along the particle-substrate interface is known, the kinetics of particle movements can be determined by coupling surface and interface diffusion fluxes. We illustrated the developed general theory by analyzing the particle movements on the surface step, on the sinusoidal substrate, and within the corner formed by two intersecting planar surfaces. We also show that the developed theory can be applied to the analysis of burrowing of rigid ceramic nanoparticles into the metal substrate.