We investigate the effect of long-range diffusive mixing on stochastic processes taking place on low-dimensional catalytic supports. As a working example, the cyclic lattice Lotka-Volterra (LLV) model is used which is conservative at the mean-field level and demonstrates fractal patterns and local oscillations when realized on low-dimensional lattice supports. We show that the local oscillations are synchronized when a weak, long-range, diffusive process is added to LLV and global oscillations of limit cycle type emerge. This phenomenon is demonstrated as a nonequilibrium phase transition and takes place when the mixing-to-reaction rate p (order parameter) is above a critical point p_{c} . The value of the critical point is shown to depend on the kinetic parameters. The global oscillations in this case emerge as a result of phase synchronization between local oscillations on sublattices.