Surface electromyogram (sEMG) has been widely used in hand gesture recognition. However, most previous studies focused on user-personalized models, which require a great amount of data from each new target user to learn the user-specific EMG patterns. In this work, we present a novel real-time gesture recognition framework based on multi-source domain adaptation, which learns extra knowledge from the data of other users, thereby reducing the data collection burdens on the target user. Additionally, compared with conventional domain adaptation methods which treat data from all users in the source domain as a whole, the proposed multi-source method treat data from different users as multiple separate source domains. Therefore, more detailed statistical information on the data distribution from each user can be learned effectively. High-density sEMG (256 channels) from 20 subjects was used to validate the proposed method. Importantly, we evaluated our method with a simulated real-time processing pipeline on continuous sEMG data stream, rather than well-segmented data. The false alarm rate during rest periods in an EMG data stream, which is typically neglected by previous studies performing offline analyses, was also considered. Our results showed that, with only 1 s sEMG data per gesture from the new user, the 10-gesture classification accuracy reached 87.66 % but the false alarm rate was reduced to 1.95 %. Our method can reduce the frustratingly heavy data collection burdens on each new user.