Multi-agent debate - multiple instances of large language models discussing problems in turn-based interaction - has shown promise for solving knowledge and reasoning tasks. However, these methods show limitations when solving complex problems that require longer reasoning chains. We analyze how multi-agent debate drifts away from the initial problem over multiple turns, thus harming task performance. We define this phenomenon as problem drift and quantify its presence across ten tasks (i.e., three generative, three knowledge, three reasoning, and one instruction-following task). We find that generative tasks drift often due to the subjectivity of the answer space (76-89%), compared to high-complexity tasks (7-21%). To identify the reasons, eight human experts analyze 170 multi-agent debates suffering from problem drift. We find the most common issues related to this drift are the lack of progress (35% of cases), low-quality feedback (26% of cases), and a lack of clarity (25% of cases). We propose DRIFTJudge, an LLM-as-a-judge method, as a first baseline to detect problem drift. We also propose DRIFTPolicy, which mitigates 31% of problem drift cases. Our study is a step toward understanding a key limitation of multi-agent debate, highlighting why longer debates can harm task performance and how problem drift could be addressed.