The Dragonfly Galaxy (MRC0152-209), at redshift z~2, is one of the most vigorously star-forming radio galaxies in the Universe. What triggered its activity? We present ALMA Cycle 2 observations of cold molecular CO(6-5) gas and dust, which reveal that this is likely a gas-rich triple merger. It consists of a close double nucleus (separation ~4 kpc) and a weak CO-emitter at ~10 kpc distance, all of which have counterparts in HST/NICMOS imagery. The hyper-luminous starburst and powerful radio-AGN were triggered at this precoalescent stage of the merger. The CO(6-5) traces dense molecular gas in the central region, and complements existing CO(1-0) data, which revealed more widespread tidal debris of cold gas. We also find ~10$^{10}$ M(sun) of molecular gas with enhanced excitation at the highest velocities. At least 20-50% of this high-excitation, high-velocity gas shows kinematics that suggests it is being displaced and redistributed within the merger, although with line-of-sight velocities of |v| < 500 km/s, this gas will probably not escape the system. The processes that drive the redistribution of cold gas are likely related to either the gravitational interaction between two kpc-scale discs, or starburst/AGN-driven outflows. We estimate that the rate at which the molecular gas is redistributed is at least ~1200 +- 500 M(sun)/yr, and could perhaps even approach the star formation rate of ~3000 +- 800 M(sun)/yr. The fact that the gas depletion and gas redistribution timescales are similar implies that dynamical processes can be important in the evolution of massive high-z galaxies.