Large Language Models (LLMs) are increasingly being deployed in applications such as chatbots, code editors, and conversational agents. A key feature of LLMs is their ability to engage in multi-turn interactions with humans or external tools, enabling a wide range of tasks. Each new request in a multi-turn interaction depends on the intermediate state, specifically the key-value (K,V) caches, from previous requests in the ongoing interaction. Existing serving engines either recompute the K,V caches or offload them to main memory. Profiling reveals that recomputation can result in over 99% of processed tokens being redundant. On the other hand, offloading K,V caches from GPU memory makes inference serving stateful, leading to load imbalances across the cluster. To address these challenges, we developed SYMPHONY. SYMPHONY leverages the observation that multi-turn work loads provide additional hints that allow K,V caches to be migrated off the critical serving path. By utilizing these hints, SYMPHONY dynamically migrates K,V caches to enable finegrained scheduling of inference requests. Our experiments demonstrate that SYMPHONY can handle over 8x the number of requests compared to state-of-the-art baselines, with a similar latency profile.