With the advancement of automated software engineering, research focus is increasingly shifting toward practical tasks reflecting the day-to-day work of software engineers. Among these tasks, software migration, a critical process of adapting code to evolving environments, has been largely overlooked. In this study, we introduce TimeMachine-bench, a benchmark designed to evaluate software migration in real-world Python projects. Our benchmark consists of GitHub repositories whose tests begin to fail in response to dependency updates. The construction process is fully automated, enabling live updates of the benchmark. Furthermore, we curated a human-verified subset to ensure problem solvability. We evaluated agent-based baselines built on top of 11 models, including both strong open-weight and state-of-the-art LLMs on this verified subset. Our results indicated that, while LLMs show some promise for migration tasks, they continue to face substantial reliability challenges, including spurious solutions that exploit low test coverage and unnecessary edits stemming from suboptimal tool-use strategies. Our dataset and implementation are available at https://github.com/tohoku-nlp/timemachine-bench.