We examined the dynamics of the deformation and phase separation of two-component vesicles. First, we numerically investigated the effects of (i) thermal noise, (ii) hydrodynamic flow induced by the line tension of the domain boundary and (iii) composition-dependent bending rigidity on the coarsening dynamics of a phase-separated pattern on the surfaces of vesicles with fixed shapes. The dynamical exponent z (N(DB) ∼ t(-z), the total length of the domain boundaries) of the coarsening of the phase-separated pattern was found to decrease from z = 1/3 under no thermal noise to 1/5 < z < 1/4 when including the effects of thermal noise. We also found that the hydrodynamic effect enhances the coarsening in a bicontinuous phase separation for a spherical vesicle. In phase separations of a shape-fixed tubular vesicle, a band-like phase separation with periodicity along the longer axis of the tube occurs because of the composition-dependent bending rigidity and the higher curvatures at the tube end-caps. Second, we also explored the dynamics of shape deformation coupled with phase separation through the bending rigidity of the membrane which depends on the local composition in lipids and found that the composition-dependent bending rigidity crucially influences the phase separation and deformation of the vesicle. The results of simulations are in good agreement with experimentally observed behavior known as 'shape convergence' (Yanagisawa et al 2008 Phys. Rev. Lett. 100 148102).