Modern AI agents increasingly combine conversational interaction with autonomous task execution, such as coding and web research, raising a natural question: What happens when an agent engaged in long-horizon tasks is exposed to user persuasion? Yet studying this possibility is challenging because long-running agent behavior is noisy and costly to reproduce, and it remains unclear which unique challenges emerge only in extended task execution. We study how belief-level intervention can influence downstream task behavior, a phenomenon we name persuasion propagation. We introduce a behavior-centered evaluation framework that distinguishes between persuasion applied during or prior to task execution. Across web research and coding tasks, we find that on-the-fly persuasion induces weak and inconsistent behavioral effects. In contrast, when the belief state is explicitly specified at task time, belief-prefilled agents conduct on average 26.9% fewer searches and visit 16.9% fewer unique sources than neutral-prefilled agents. These results suggest that persuasion, even in prior interaction, can affect the agent's behavior, motivating behavior-level evaluation in agentic systems.