A common solution for mitigating outdated or incorrect information in Large Language Models (LLMs) is to provide updated facts in-context or through knowledge editing. However, these methods introduce knowledge conflicts when the knowledge update fails to overwrite the model's parametric knowledge, which propagate to faulty reasoning. Current benchmarks for this problem, however, largely focus only on single knowledge updates and fact recall without evaluating how these updates affect downstream reasoning. In this work, we introduce TRACK (Testing Reasoning Amid Conflicting Knowledge), a new benchmark for studying how LLMs propagate new knowledge through multi-step reasoning when it conflicts with the model's initial parametric knowledge. Spanning three reasoning-intensive scenarios (WIKI, CODE, and MATH), TRACK introduces multiple, realistic conflicts to mirror real-world complexity. Our results on TRACK reveal that providing updated facts to models for reasoning can worsen performance compared to providing no updated facts to a model, and that this performance degradation exacerbates as more updated facts are provided. We show this failure stems from both inability to faithfully integrate updated facts, but also flawed reasoning even when knowledge is integrated. TRACK provides a rigorous new benchmark to measure and guide future progress on propagating conflicting knowledge in multi-step reasoning.