Session-based recommendation (SBR) and multi-behavior recommendation (MBR) are both important problems and have attracted the attention of many researchers and practitioners. Different from SBR that solely uses one single type of behavior sequences and MBR that neglects sequential dynamics, heterogeneous SBR (HSBR) that exploits different types of behavioral information (e.g., examinations like clicks or browses, purchases, adds-to-carts and adds-to-favorites) in sequences is more consistent with real-world recommendation scenarios, but it is rarely studied. Early efforts towards HSBR focus on distinguishing different types of behaviors or exploiting homogeneous behavior transitions in a sequence with the same type of behaviors. However, all the existing solutions for HSBR do not exploit the rich heterogeneous behavior transitions in an explicit way and thus may fail to capture the semantic relations between different types of behaviors. However, all the existing solutions for HSBR do not model the rich heterogeneous behavior transitions in the form of graphs and thus may fail to capture the semantic relations between different types of behaviors. The limitation hinders the development of HSBR and results in unsatisfactory performance. As a response, we propose a novel behavior-aware graph neural network (BGNN) for HSBR. Our BGNN adopts a dual-channel learning strategy for differentiated modeling of two different types of behavior sequences in a session. Moreover, our BGNN integrates the information of both homogeneous behavior transitions and heterogeneous behavior transitions in a unified way. We then conduct extensive empirical studies on three real-world datasets, and find that our BGNN outperforms the best baseline by 21.87%, 18.49%, and 37.16% on average correspondingly. A series of further experiments and visualization studies demonstrate the rationality and effectiveness of our BGNN. An exploratory study on extending our BGNN to handle more than two types of behaviors show that our BGNN can easily and effectively be extended to multi-behavior scenarios.