We investigate the dynamics of phase separation processes in the presence of added surfactant molecules with an asymmetric shape, e.g., surfactant molecules with asymmetric hydrophilic/hydrophobic sizes, or block copolymers with asymmetric compositions of subchains. Upon adsorption onto interfaces, such asymmetric surfactant molecules produce a spontaneous curvature of interfaces. The analysis is based on the hybrid model, with which one can investigate the macroscopic phase separation phenomena while keeping some essential molecular features of the surfactant. Computer simulations are performed using the hybrid model and we found a spontaneous morphological change from a bicontinuous structure to a globular structure during the phase separation process. Such a morphological change is caused by the spontaneous curvature of the interfaces where the surfactant molecules are adsorbed. A simple argument to explain such a phenomenon is also presented.