We propose Flow Mismatching, an unsupervised anomaly detection method that deliberately avoids reconstruction-based paradigms. Instead, we treat flow matching as geometric dynamics and leverage a key insight: anomalies occur at places where the learned normal flow disagrees with the geometric path toward a test image. Given a flow matching model trained only on normal images, we probe its learned velocity field along affine paths from Gaussian noise to a target image. Along each path, we compare the model-predicted velocity, which follows normal generative dynamics, with the geometric velocity toward the target, which includes any anomalous content. Anomalies induce strong local disagreement between these velocities. Aggregating the mismatch over different time steps and multiple paths yields pixel-wise heatmaps and image-level scores without test-time optimization, feature memories, or additional calibration. Our analysis shows that the population mismatch decomposes into an irreducible denoising term and a Fisher-divergence term between the test-path and normal-path score functions, which identifies the score-gap component that drives anomaly separation and explains the effectiveness of robust path aggregation. Extensive experiments on MVTec-AD and VisA demonstrate superior performance compared with SOTA reconstruction-based and recent flow matching-based approaches.