We explore active audio-visual separation for dynamic sound sources, where an embodied agent moves intelligently in a 3D environment to continuously isolate the time-varying audio stream being emitted by an object of interest. The agent hears a mixed stream of multiple audio sources (e.g., multiple people conversing and a band playing music at a noisy party). Given a limited time budget, it needs to extract the target sound accurately at every step using egocentric audio-visual observations. We propose a reinforcement learning agent equipped with a novel transformer memory that learns motion policies to control its camera and microphone to recover the dynamic target audio, using self-attention to make high-quality estimates for current timesteps and also simultaneously improve its past estimates. Using highly realistic acoustic SoundSpaces simulations in real-world scanned Matterport3D environments, we show that our model is able to learn efficient behavior to carry out continuous separation of a dynamic audio target. Project: https://vision.cs.utexas.edu/projects/active-av-dynamic-separation/.