Partially engulfed compound droplets are present in many natural and technological processes in which heat is exchanged or solutes diffuse between the various immiscible phases through the interfaces. The variations of the surface tension on the multiple interfaces induce a Marangoni type motion in and around such drops. Two types of thermal fields and the resulting dynamics are reported here. Flow induced by an external field and a spontaneous thermocapillary motion induced by a thermodynamics state not at equilibrium and the geometric non-uniformity of the system. The motion under external forcing is, in general, induced in the direction of the imposed gradient. However, under certain physical conditions and configuration of the drop, an unexpected motion against the gradient is observed. The thermocapillary migration velocity achieves various maxima under specific conditions and recovers values known for single phase drops at asymptotic limits. We present results for a variety of parameters such as configuration of the compound drop, viscosity, thermal conductivity ratio, the dependence of the various interfacial tensions on temperature and the volume ratio of the phases within the drop.