Phase-detection focus is the current speed champ and the gap is closing quickly, but it's not there yet. The three new Olympus µ4/3 bodies added parallel CDAF processors that take them closer to real time focus with compatible lenses. Where they fall flat is running dlsr lenses designed for PDAF, where the response is poky.
The driving force here is video, as DSLR video is comical until the camera is rigged up with cine gear to pull focus and run the aperture in real time. This transformation makes them broadcast and even cinema quality, but irrelevant for backpackers. The mirrorless cameras can do these things out of the box, with no rigging required, and PDAF enables them to follow focus in real time (with system lenses). This is their hole card, at present, for folks interested primarily in video.
Fast forward a few years and dslrs will be photographic dinosaurs, but not yet. In the meantime, anybody upgrading from their P&S or phonecam will be torn which direction to go: mirrorless or dslr. The answer can only come from a careful appraisal of your projected uses. It's easy to get distracted by other folks' goals, or unrealistic goals.