We design and implement Megaphone, a data migration mechanism for stateful distributed dataflow engines with latency objectives. When compared to existing migration mechanisms, Megaphone has the following differentiating characteristics: (i) migrations can be subdivided to a configurable granularity to avoid latency spikes, and (ii) migrations can be prepared ahead of time to avoid runtime coordination. Megaphone is implemented as a library on an unmodified timely dataflow implementation, and provides an operator interface compatible with its existing APIs. We evaluate Megaphone on established benchmarks with varying amounts of state and observe that compared to naïve approaches Megaphone reduces service latencies during reconfiguration by orders of magnitude without significantly increasing steady-state overhead.