The improvement of web agents on current benchmarks raises the question: Do today's agents perform just as well when the web changes? We introduce TimeWarp, a benchmark that emulates the evolving web using containerized environments that vary in UI, design, and layout. TimeWarp consists of three web environments, each with six UI versions spanning different eras of the internet, paired with a set of complex, realistic tasks requiring different forms of web navigation. Our experiments reveal web agents' vulnerability to changes and the limitations of behavior cloning (BC) on single-version trajectories. To address this, we propose TimeTraj, a simple yet effective algorithm that uses plan distillation to collect trajectories across multiple versions. By training agents on teacher rollouts using our BC-variant, we achieve substantial performance gains: $20.4\%\rightarrow37.7\%$ for Qwen-3 4B and $0\%\rightarrow27.0\%$ for Llama-3.1 8B models. We hope our work helps researchers study generalization across web designs and unlock a new paradigm for collecting plans rather than trajectories, thereby improving the robustness of web agents.