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The deformation dynamics of a moving compound droplet inside a microcapillary tube is investigated in Newtonian and non-Newtonian ambient fluids. The numerical method used is a ternary phase-field model to determine the temporal evolution and the subsequent hydrodynamics of compound droplets traveling inside a microcapillary tube. By solving the axisymmetric momentum equations, the compound droplet movement is identified by tracing the spatiotemporal evolution of a pair of two deformable interfaces. Results indicate that the rheological properties of the continuous phase significantly influence the compound droplet characteristics. The position of the center of the mass of both inner and outer drops along the tube axis increases linearly in a Newtonian ambient fluid while it is nonlinear in a non-Newtonian medium, and the deviation from the linear behavior increases as the concentration of the polymer …