Using the front-tracking method, we numerically analyze the migration and shape deformation of a multi-core compound droplet through a constricted micro-channel. Due to the interaction with the constriction structure, the velocity field of the flow changes, significantly affecting the motion dynamics of the multi-core compound droplet. Two regimes are identified: the merging mode (M), where the core droplets coalesce to form a single-core compound droplet, and the non-merging mode (NM). The effects of the Weber number (We) and constriction geometry (constriction length HCon/Rshell) on droplet dynamics are analyzed. The study reveals that at low Reynolds number within the investigated range, the droplet migrates through the constricted region in the merging regime, where the internal droplets coalesce. Variations in other parameters, including the We, the HCon/Rshell lead to transitions between merging (M) and non-merging (NM) migration modes.