Particles confined in droplets are called compound particles. They are encountered in various biological and soft matter systems. Hydrodynamics can play a decisive role in determining the configuration and stability of these multiphase structures during their preparation and use. Therefore, we investigate the dynamics and stability of a concentric compound particle under external forces and imposed flows. The governing equations are solved analytically in the inertia-less limit using the standard technique of superposition of vector harmonics and the solutions obtained are reported in terms of steady state flow fields, the viscous drag on the particle and the time evolution of the confining drop shape. The limiting form of a compound particle as a thin film coated rigid particle is analyzed in each case. We find that the concentric configuration of a rotating compound particle is a steady state solution, and we calculate the extra force required to stabilize the concentric configuration of a translating compound particle. A comprehensive comparison of drop deformations in various linear ambient flows is also provided. Based on the findings, we propose pulsatile flow as a reliable method to transport compound particles without breakup of the confining drop. Thus, our analysis provides useful guidelines for preparation and transportation of stable compound particles in the context of nucleated cells, aerosols, droplet-based encapsulation of motile organisms and polymer microcapsules. © 2019 The Royal Society of Chemistry.