Recent approaches to point tracking are able to recover the trajectory of any scene point through a large portion of a video despite the presence of occlusions.They are, however, too slow in practice to track every point observed in a single frame in a reasonable amount of time.This paper introduces DOT, a novel, simple and efficient method for solving this problem.It first extracts a small set of tracks from key regions at motion boundaries using an off-the-shelf point tracking algorithm.Given source and target frames, DOT then computes rough initial estimates of a dense flow field and visibility mask through nearest-neighbor interpolation, before refining them using a learnable optical flow estimator that explicitly handles occlusions and can be trained on synthetic data with ground-truth correspondences.We show that DOT is significantly more accurate than current optical flow techniques, outperforms sophisticated ``universal'' trackers like OmniMotion, and is on par with, or better than, the best point tracking algorithms like CoTracker while being at least two orders of magnitude faster.Quantitative and qualitative experiments with synthetic and real videos validate the promise of the proposed approach.