Portrait video editing has attracted wide attention thanks to its practical applications. Existing methods either target fixed-length clips or perform temporally inconsistent per-frame editing. In this work, we present a brand new system, StreamEdit, which is primarily designed to edit streaming videos. Our system follows the ideology of editing propagation to ensure temporal consistency. Concretely, we choose to edit only one reference frame and warp the outcome to obtain the editing results of other frames. For this purpose, we employ a warping module, aided by a probabilistic pixel correspondence estimation network, to help establish the pixel-wise mapping between two frames. However, such a pipeline requires the reference frame to contain all contents appearing in the video, which is scarcely possible especially when there exist large motions and occlusions. To address this challenge, we propose to adaptively replace the reference frame, benefiting from a heuristic strategy referring to the overall pixel mapping uncertainty. That way, we can easily align the editing of the before- and after-replacement reference frames via image inpainting. Extensive experimental results demonstrate the effectiveness and generalizability of our approach in editing streaming portrait videos. Code will be made public.