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Long-range interactions are shown to cause, as time evolves, consecutive reversals of directed currents for dilute ensembles of particles in driven lattices. These current reversals are based on a general mechanism that leads to an interaction-induced accumulation of particles in the regular regions of the underlying single-particle phase space and to a synchronized single-particle motion as well as enhanced efficiency of Hamiltonian ratchets. Suggestions for experimental implementations using ionized mesoscopic clusters in micromechanical lattices or dipolarly interacting colloidal particles in ac-driven optical lattices are provided.