Autofocus during video is still an important feature. That, and continuous autofocus with moving subjects, have been problem areas for Micro Four Thirds. High end DSLR cameras handle continuous autofocus, e.g., for sports and wildlife, very good. This is due to using PDAF technology. Micro Four Thirds cameras, with the exception of the Olympus E-M1 so far, only use CDAF, not PDAF.
Achieving efficient autofocus during video with CDAF is a matter of having fast image processing capabilites, combined with good algorithms for interpreting the data. GH3, being the newest and most powerful of the GH-line, of course has the best potential here. The Lumix GM1 is also a premium camera, however, it does not have the same video features as the GH3. For example, it lacks the MOV video format, has lower maximum bitrate, and cannot record progressive video at 50 FPS (PAL regions) or 60 FPS (NTSC regions) in 1920x1080 mode.
Here is a video comparison of the GH3 and GM1 both doing AF with the Lumix G 14mm f/2.5 pancake lens mounted.
Both cameras were mounted to a Desmond Mini Dual Camera Bracket, typically used for stereo photography. On the lenses, I have used 46mm to 37mm step down rings as hoods. They do a good job of keeping the front lens elements safe from accidents, in my opinion, while also keeping out some stray light. If you want to use them, you also need a 37mm front lens cap.
Both cameras were set to use the max aperture, f/2.5, 1/50s shutter speed, and 25 frames per second in 1080p mode. Since the scenes were rather dark, the ISO was set to around 4000. Here are the results:
Despite what one would probably guess, the GH3 is not universally the best performer here. It probably gets the focus right somewhat faster on average, but the GM1 also does remarkably well. It shows that Panasonic's autofocus technology is advancing, and also trickling down to the smaller models.
The first generations of Panasonic cameras would show typical CDAF behaviour when focusing, i.e., jogging the focus back and forth once to confirm the focus correctness. This can be quite annoying. It is very pleasing to see that the Lumix GM1 avoids this behaviour. Here you can see that even the GH2, the previous top dog, has the annoying back-and-forth-focus when changing the focus distance. So Panasonic have come a long way the last couple of years.
Even if you might not buy the Lumix GM1 specifically for video use, due to the less than optimal ergonomics, you can still trust it to do a good job if you want to test the video function.
For dedicated video use, I would look for a camera with a tiltable LCD viewfinder, or an eye level viewfinder. Not having any of these means that you need to hold the camera out in front of you while filming, which makes it very hard to hold it stably. With a tilt viewfinder, you can rest the camera against your waist, and look down into the LCD screen for framing.
Both cameras were set to auto white balance (AWB), and it appears that the GH3 gives the most pleasing results in terms of colours. Both cameras had the same image parameters, but still the GM1 gives more saturated colours, and warmer colours.
Despite using very high ISO (around 4000), both camera perform very well here. It shows that not only the flagship camera in terms of video cam be trusted to perform well in this area: Even the GM1 appears to get video right.