Human conversations contain natural and reasonable topic shifts, reflected as the concept flows across utterances. Previous researches prove that explicitly modeling concept flows with a large commonsense knowledge graph effectively improves response quality. However, we argue that there exists a gap between the knowledge graph and the conversation. The knowledge graph has limited commonsense knowledge and ignores the characteristics of natural conversations. Thus, many concepts and relations in conversations are not included. To bridge this gap, we propose to enhance dialogue generation with conversational concept flows. Specifically, we extract abundant concepts and relations from natural conversations and build a new conversation-aware knowledge graph. In addition, we design a novel relation-aware graph encoder to capture the concept flows guided by the knowledge graph. Experimental results on the large-scale Reddit conversation dataset indicate that our method performs better than strong baselines, andfurther analysis verifies the effectiveness of each component. All our code and data will be publicly available after acceptance.