In this paper, we tackle the problem of performing consistent, unified modifications to a multi-shot video sequence. This task is particularly challenging because multi-shot videos consist of discontinuous temporal segments that vary significantly in viewpoint, camera scale, and subject pose, leading to severe identity drift and cumulative error propagation. Achieving coherent edits requires establishing reliable cross-shot semantic awareness to maintain stable subject appearance and visual continuity across these disjointed boundaries. To address this, we propose MSEditor, the first framework designed specifically for consistent multi-shot video editing. To overcome the scarcity of high-quality multi-shot training data, we repurpose existing multi-view video datasets to provide robust cross-shot supervision. Architecturally, we introduce a Supervisory Adapter that injects this cross-shot information into the diffusion backbone, enabling the model to learn identity-consistent representations. Furthermore, to effectively mitigate cumulative errors and ensure long-range temporal coherence, we design a Cross-Shot Packing strategy that dynamically aggregates information from semantically related shots within the self-attention window. Extensive experiments demonstrate that MSEditor significantly outperforms existing methods on our curated multi-shot video editing benchmark in terms of identity preservation, temporal stability, and overall visual quality.