Streaming session-based recommendation (SSR) is a challenging task that requires the recommender system to do the session-based recommendation (SR) in the streaming scenario. In the real-world applications of e-commerce and social media, a sequence of user-item interactions generated within a certain period are grouped as a session, and these sessions consecutively arrive in the form of streams. Most of the recent SR research has focused on the static setting where the training data is first acquired and then used to train a session-based recommender model. They need several epochs of training over the whole dataset, which is infeasible in the streaming setting. Besides, they can hardly well capture long-term user interests because of the neglect or the simple usage of the user information. Although some streaming recommendation strategies have been proposed recently, they are designed for streams of individual interactions rather than streams of sessions. In this paper, we propose a Global Attributed Graph (GAG) neural network model with a Wasserstein reservoir for the SSR problem. On one hand, when a new session arrives, a session graph with a global attribute is constructed based on the current session and its associate user. Thus, the GAG can take both the global attribute and the current session into consideration to learn more comprehensive representations of the session and the user, yielding a better performance in the recommendation. On the other hand, for the adaptation to the streaming session scenario, a Wasserstein reservoir is proposed to help preserve a representative sketch of the historical data. Extensive experiments on two real-world datasets have been conducted to verify the superiority of the GAG model compared with the state-of-the-art methods.