The dynamic response of a raft-exhibiting giant liposome to external stimuli, such as the addition of Triton X-100 or osmotic stress, was studied. We observed that daughter vesicles are generated inside of the liposome through endocytic budding. It was found that the budding to generate daughter vesicles is classified into two different routes, simple budding through the invagination of a whole raft and budding from the boundary of a raft accompanied by waving motion. Smaller rafts show a preference for simple budding, whereas large rafts mainly adopt the other process. We discuss the mechanism of this difference in terms of the kinetic pathway of internalization by considering the line energy and bending energy of the membrane.