In this paper we study functional limits of occupation time fluctuations of a kind of generalized branching particle systems, whose branching laws depend on the particles' sites, in the case of intermediate dimensions. The results show that the limit processes depend on the behaviors of branching laws. When the variance function of branching laws slowly converges to 0, the limit processes have
... [Show full abstract] complicate temporal but simple spatial structures and the branching laws of particles nearby infinity rather than original site affect the norming parameters and the temporal structures of limit processes. When the variance function of branching laws is integrable, the limit processes have simple temporal but complicate spatial structures and the jumping phenomenon in classical branching systems at critical dimensions disappears from the norming parameters and the structures of limit processes.