Multiphase flows abound in nature and enterprises. Our daily interactions with fluids - washing, drinking, and cooking, for example - occur at a free surface and within the realm of multiphase flows. The applications of multiphase flows within the context of emulsions, which are caused by mixing two immiscible fluids, have been of interest since the nineteenth century: compartmentalizing one fluid in another is particularly of interest in applications in pharmaceutical, materials, microfluidics, chemical, and biological engineering. Even more control in compartmentalization and delivery can be obtained through the usage of double emulsions, which are emulsions of smaller drops (i.e., inner drop) within larger drops (i.e., outer drop). The goal of this work is to understand the dynamic behavior of compound drops in confined flow at low Reynolds numbers. These behaviors include the migration patterns, limit cycles, and equilibrium locations in confined flows such as channel flows.<br> <br>Firstly, we look at non-concentric compound drops that are subject to simple shear flows. The eccentricity in the inner drop is either within the place of shear, normal to the plane of shear, or mixed. We show unreported motions that persist throughout time regardless of the initial eccentricity, given that the deformations of the inner and outer drops are small. Understanding the temporal dynamics of compound drops within the simple shear flow, one of the simplest background flows that may be imposed, allows us to probe at the dynamics of more complicated background flows.<br> <br>Secondly, we look at the lateral migration of compound drops in a Poiseuille flow. Depending on the initial condition, we show that there are multiple equilibria. We also show that the majority of initial configurations results in the compound drop with symmetry about the short wall direction. We then show the time it takes for the interfaces to merge if a given initial configuration does not reach the aforementioned symmetry.<br> <br>Thirdly, while the different equilibria of compound drops offer some positional differences at different radii ratio, we show that the lift force profiles at non-equilibrium locations offer distinctly different results for compound drops with different radii ratio. We then look at how this effect is greater than changes that arise due to viscosity ratio changes, and offer insights on what may create such a change in the lift force profile.