A dynamically tunable multiband plasmon-induced transparency (PIT) effect in a series of rectangle cavities coupled with a graphene nanoribbon waveguide system is investigated theoretically and numerically by tuning the Fermi level of the graphene rectangle cavity. A single-PIT effect is realized using two different methods: one is the direct destructive interference between bright and dark modes, and the other is the indirect coupling through a graphene nanoribbon waveguide. Moreover, dual-PIT effect is obtained by three rectangle cavities side-coupled with a graphene nanoribbon waveguide. Results show that the magnitude of the dual-PIT window can be controlled between 0.21 and 0.74, and the corresponding group index is controlled between 143.2 and 108.6. Furthermore, the triple-PIT effect is achieved by the combination of bright–dark mode coupling and the cavities side-coupled with waveguide mechanism. Thus, sharp PIT windows can be formed, a high transmission is maintained between 0.51 and 0.74, and the corresponding group index is controlled between 161.4 and 115.8. Compared with previously proposed graphene-based PIT effects, the size of the introduced structure is less than 0.5 μm2. Particularly, the slow light effect is crucial in the current research. Therefore, a novel approach is introduced toward the realization of optical sensors, optical filters, and slow light and light storage devices with ultra-compact, multiband, and dynamic tunable.