We introduce a one-dimensional model involving the nucleation and the drift of many particles. The model originates from interacting kink systems and simulates time evolution in modulated systems. In this model the nucleation rate of a particle depends nonlocally on the density of preexisting particles and the drift of particles is due to a weak and repulsive interaction among them. We first study the statistics of this model in the case that the drift of particles is negligible, and then consider the effects of the drift of particles.