According to Olympus' published specs the E-M1 can track focus on a moving object while shooting at 6.5 fps, with a buffer of 60 frames when shooting RAW. At this speed it also gives a live view feed briefly between frames, which makes it easy to pan the camera to keep the subject in the frame, and overall makes the shooting experience very similar to using an SLR. If you don't need focus tracking it can shoot even faster, at up to 10 fps.
Continuous and tracking AF has been a stumbling block for mirrorless cameras, so far, because contrast-detection AF can't give the camera information about where to focus the next shot. Only Nikon's 1 System cameras, with their on-chip phase detect AF, have bucked this trend up until now. Our initial impression is that the E-M1 does similarly well, making it is a big step forward for Olympus. It does a good job of tracking a target and getting a healthy proportion of shots in focus.
The examples below were shot at a press event hosted by Olympus at Castle Leslie in Ireland. In the first set the horse is trotting briskly through the water towards the camera, and in the second it's galloping up a hill. The lens used was the M.Zuiko Digital ED 60mm 1:2.8 Macro, and all of the images were shot at F2.8. We've shown 100% crops from selected frames beneath the main rollovers, but you can click on any thumbnail to download a full size image and check focus for yourself. In both cases the camera has done an impressive job of maintaining focus on the horse, presumably helped by its on-chip PDAF.