Existing methods on audio-visual deepfake detection mainly focus on high-level features for modeling inconsistencies between audio and visual data. As a result, these approaches usually overlook finer audio-visual artifacts, which are inherent to deepfakes. Herein, we propose the introduction of fine-grained mechanisms for detecting subtle artifacts in both spatial and temporal domains. First, we introduce a local audio-visual model capable of capturing small spatial regions that are prone to inconsistencies with audio. For that purpose, a fine-grained mechanism based on a spatially-local distance coupled with an attention module is adopted. Second, we introduce a temporally-local pseudo-fake augmentation to include samples incorporating subtle temporal inconsistencies in our training set. Experiments on the DFDC and the FakeAVCeleb datasets demonstrate the superiority of the proposed method in terms of generalization as compared to the state-of-the-art under both in-dataset and cross-dataset settings.