To harness the benefits of both reconfigurable intelligent surface (RIS) and dynamic metasurface antenna (DMA), we consider the hybrid RIS and DMA assisted multiple-input multiple-output (MIMO) multiple-access channel (MAC) over spatially correlated Rician fading, in which multiple multi-antenna users send the transmitted signals to the DMA-based base station (BS) with the assistance of a RIS. The objective is to maximize the achievable ergodic sum-rate by jointly designing the transmit covariance matrix of users, the phase shift matrix of RIS, and the DMA weight matrix at BS only with statistical channel state information. By capitalizing on large random matrix theory, a closed-form asymptotic ergodic sum-rate is first obtained. Then, we propose a modified water-filling algorithm to design the optimal transmit covariance matrix under the power consumption and specific absorption rate constraints. Next, we design the phase shift matrix of RIS via the projected gradient ascent algorithm, subject to the non-convex unit-modular constraint. To find the constrained DMA weight matrix, we further resort to the optimal solution of the unconstrained DMA problem and adopt the alternating optimization method. The proposed algorithm is numerically shown to improve the sum-rate compared to the baseline schemes, verifying the effectiveness of the proposed schemes.