Reconfigurable intelligent surfaces (RISs) have emerged as a candidate technology in future wireless communications for their appealing advantages in controlling the direction of the beam to improve the wireless communication performance. In addition, as a new paradigm to realize large-scale antenna arrays with reduced cost and power consumption, dynamic metasurface antennas (DMAs) have great potential to be deployed at the base station (BS) as a low-cost alternative for traditional antenna array. For the best mutual benefit of RIS and DMA technologies, we amalgamate them to investigate the hybrid RIS and DMA assisted MIMO interfering broadcast channel with high communication sum-rate at low implementation cost, wherein a DMA-based BS communicates with multiple multi-antenna users via the assistance of RIS. Due to the physical implementation of RIS and DMA, conventional MIMO interfering broadcast channel models are no longer valid with additional phase shifts of all RIS reflecting elements and Lorentzian response of the metamaterial elements inside the DMA microstrip. Focusing on the weighted sum-rate performance of the system with additional interference-plus-noise among users, we propose an efficient mechanism to jointly design the transmit precoding matrices at the BS, the phase shift matrix at the RIS, and the weight matrix at the DMA. The optimal design is non-trivial owing to the inter-user interference, not to mention the unique constraints on RIS reflecting elements and DMA weight matrix. By transforming the original weighted sum-rate maximization problem to a solvable one, we propose an alternating algorithm to tackle these issues, where the three optimization variables are all obtained in closed-form expressions. Simulation results verify the superior weighted sum-rate performance of the proposed algorithms. We also find that under the considered framework the performance loss brought by the mutual coupling in the DMA arrays is negligible compared to the conventional array antennas.