We have investigated the dynamics of a simple and a compound droplet separately passing through a two-dimensional Poiseuille channel flow by using the volume-of-fluids method under low Reynolds numbers (Re=1). A simple droplet while passing through the channel shows an elongation in shape in the streamwise direction and compression in the transverse direction. A round symmetric droplet evolves into a pear-like shape with time, if it is released at an off-centre to the channel mid-line. If a compound droplet (which consists of a thin shell separating an inner fluid zone from an outer fluid zone) is released in the channel either at the mid-line or with an offset, we observed the round shape evolves into a slipper-like shape. The inner droplet breaks away from the outer one within a short period. A detailed study of droplet deformation and its breakup has been carried out for different fluid properties and the compound droplet size ratios, and we found the lateral migration is high for the compound droplets when compared to the simple droplets. We also observed that the break time of the compound droplet decreases with the increase in the offset release location exponentially.