Existing fake information detection methods primarily focus on analyzing textual and structural features, failing to comprehensively consider subtle clues in the propagation process of misinformation (e.g., users supporting articles and posts about articles). Evidence-based reasoning methods can achieve fine-grained modeling of these subtle clues during propagation through complex reasoning techniques, thereby enabling fake information detection. However, evidence is often large-scale, making it highly challenging to perform fine-grained reasoning by considering all evidence. The logical reasoning ability of human thinking can better connect subtle clues and identify the most critical ones. A fine-grained reasoning framework, which better reflects the logical process of human thinking, is proposed to model these subtle clues, thereby shifting toward fine-grained reasoning for fake information detection. Specifically, first, a mutually reinforced claim-evidence graph is constructed based on interactions between text and users to identify key evidence. Second, a prior-aware dual-channel kernel graph network is designed to model subtle differences among evidence. To validate the effectiveness of the proposed method, experiments are conducted on two public datasets. The results demonstrate that the proposed model improves accuracy by 5% and 7.9%, respectively, compared to the best baseline methods.