We examine the interaction of a lipid bilayer membrane with a spherical particle in solution using dissipative particle dynamics, with the aim of controlling the passage of foreign objects into and out of vesicles. Parameters are chosen such that there is a favorable adhesive interaction between the membrane and the particle. Under these conditions, the membrane wraps the particle in a process resembling phagocytosis in biological cells. We find that, for a homogeneous membrane with a uniform attraction to the particle, the membrane is unable to fully wrap the particle when the adhesion strength is below a certain value. This is observed even in the limit of zero membrane tension. When the adhesion strength is increased above the threshold value, the membrane fully wraps the particle. However, the wrapped particle remains tethered to the larger membrane. We next consider an adhesive domain, or raft, in an otherwise nonadhesive membrane. We find that, when the particle is wrapped by the raft, the line tension at the raft interface promotes fission, allowing the wrapped particle to detach from the larger membrane. This mechanism could be used to allow particles to cross a vesicle membrane.