Histopathology slide digitization introduces scanner-induced domain shift that can significantly impact computational pathology models based on deep learning methods. In the state-of-the-art, this shift is often characterized at a broad scale (slide-level or dataset-level) but not patch-level, which limits our comprehension of the impact of localized tissue characteristics on the accuracy of the deep learning models. To address this challenge, we present a domain shift analysis framework based on UWarp, a novel registration tool designed to accurately align histological slides scanned under varying conditions. UWarp employs a hierarchical registration approach, combining global affine transformations with fine-grained local corrections to achieve robust tissue patch alignment. We evaluate UWarp using two private datasets, CypathLung and BosomShieldBreast, containing whole slide images scanned by multiple devices. Our experiments demonstrate that UWarp outperforms existing open-source registration methods, achieving a median target registration error (TRE) of less than 4 pixels (<1 micrometer at 40x magnification) while significantly reducing computational time. Additionally, we apply UWarp to characterize scanner-induced local domain shift in the predictions of Breast-NEOprAIdict, a deep learning model for breast cancer pathological response prediction. We find that prediction variability is strongly correlated with tissue density on a given patch. Our findings highlight the importance of localized domain shift analysis and suggest that UWarp can serve as a valuable tool for improving model robustness and domain adaptation strategies in computational pathology.