The present study analyses the global linear stability of a two-layer channel flow when a train of solid particles is flowing near the deformable liquid-liquid interface. Three different mechanisms of instability are identified: shear, interfacial and migration modes. We demonstrate that the interfacial instability, associated to the viscosity jump at the interface, is coupled to the migration of the particle. The stability of the flow configuration is assessed for different values of the governing parameters - fluids viscosity and flow rate ratios, particle position, interparticle distance and Reynolds and capillary numbers. Our numerical results are compared with those corresponding to the particle-free flow configuration to show that, if the particle is immersed in the more-viscous fluid, its effect is always destabilizing. Remarkably, for certain flow parameters the presence of the particle stabilizes the interface when it is flowing in the less-viscous liquid. The effect of the particles becomes more significant as the Ohnesorge number $Oh = \sqrt{Re/Ca}$ increases.