The shape relaxation of supported crystallites is studied by means of kinetic Monte Carlo simulations, initialising the system with different configurations. At low temperature, when the facet nucleation limits the relaxation, the simulations show that the equilibration mechanism and the equilibration time scaling laws depend strongly on the initialisation. In this regime, the adhesion strongly increases the stability of intermediate configurations with large contact area. The relationship between the different equilibration pathways and the equilibration scaling laws is discussed considering the dependence of the nucleation barrier energy G∗ on the particle energetics in the regions with the largest kinks and steps density.