The time history is examined of the motion of a compound multiphase drop formed by a vapour bubble completely covered by its liquid phase in another immiscible liquid. The compound drop is growing or collapsing owing to change of phase while it is translating under buoyant forces. In the limit of large surface-tension forces the interfaces are spherical. An exact analytical solution for the fluid-mechanical part of the problem can be obtained. The heat-transfer treatment of the problem, however, requires numerical solution if we are to include convective terms along with time dependence. The drag component induced by radial velocity contributes to the total drag on the bubble in eccentric configuration. This drag force is towards the centre of the drop in the case of growth and has an effect of restoring concentricity. However, it is found that, in the case of growth, the compound drop, in general, cannot maintain its configuration of two non-intersecting eccentric spheres. On the other hand, in the case of collapse the bubble stays inside the drop if the collapse velocity is high enough. The complete analysis exhibits some interesting flow patterns relating to compound drops and bubbles. The time-dependent Nusselt number for a single bubble generally decreases with time but it may have a strong dependence on the compound-drop configuration, as well as the conductivities of the participating liquids. The radial convection opposes heat transfer but it has to compete with translatory convection, which is usually overwhelming in the case of growth.