Solvent-based post-combustion carbon capture (PCC) technology is a promising, near-term solution for decarbonizing power generation and industrial facilities. Model-based process simulation is crucial for the optimal design and operation of the PCC process. Recently, data-driven models have gained attention due to their adaptability, efficient computation and high accuracy. However, the nonlinearity, strong couplings and multi-time scale features of the PCC process pose significant challenges for model identification. To this end, this paper proposes a multi-gate mixture-of-experts incorporating dual-stage attention-based encoder-decoder (MMoE-DAED) network for dynamic modeling of the PCC process under wide operating conditions. An encoder-decoder composed of long short-term memory (LSTM) network is employed to extract features from the time-dependent input data and learn the complex dynamic interactions caused by the inertial and delay properties of the process. Dual-stage attention mechanism is incorporated into the encoder and decoder respectively to select the most relevant input features and their correlations within the time series data. To enhance multi-output prediction accuracy, multi-gate mixture-of-experts (MMoE) framework that considers correlations of multitask learning is implemented. Simulation results using operating data from a PCC experimental setup indicate that the proposed modeling approach accurately predicts the steady-state values and dynamic trends of the CO2 capture rate and stripper bottom temperature over a wide operating range. The RMSE, MAPE and R2 indices for the CO2 capture rate are 2.1592, 0.0295, 0.9641, respectively, and for the stripper bottom temperature are 0.1491, 0.0003, 0.9833, respectively. Validations on a PCC simulator further verify the accuracy and efficiency of the MMoE-DAED model, which enables an 80.87% reduction in computation time compared to the simulator. This paper points to a new direction for the data-driven dynamic modeling of complex energy conversion processes.