In this paper, we focus on the challenge of individual variability in affective brain-computer interfaces (aBCI), which employs electroencephalogram (EEG) signals to monitor and recognize human emotional states, thereby facilitating the advancement of emotion-aware technologies. The variability in EEG data across individuals poses a significant barrier to the development of effective and widely applicable aBCI models. To tackle this issue, we propose a novel transfer learning framework called Semi-supervised Domain Adaptation with Dynamic Distribution Alignment (SDA-DDA). This approach aligns the marginal and conditional probability distribution of source and target domains using maximum mean discrepancy (MMD) and conditional maximum mean discrepancy (CMMD). We introduce a dynamic distribution alignment mechanism to adjust differences throughout training and enhance adaptation. Additionally, a pseudo-label confidence filtering module is integrated into the semi-supervised process to refine pseudo-label generation and improve the estimation of conditional distributions. Extensive experiments on EEG benchmark databases (SEED, SEED-IV and DEAP) validate the robustness and effectiveness of SDA-DDA. The results demonstrate its superiority over existing methods in emotion recognition across various scenarios, including cross-subject and cross-session conditions. This advancement enhances the generalization and accuracy of emotion recognition, potentially fostering the development of personalized aBCI applications. The source code is accessible at https://github.com/XuanSuTrum/SDA-DDA.