Hi Grant,
After my BIG DUMMY SPIT of late last year when I retired all my Olympus gear and replaced it with a Canon EOS 6D, a 24-105 lens and a 100-400 lens, I discovered the huge difference in low-light performance between the EOS 6D and my old Olympus E-3. For me, it really was a huge improvement.
But then I noticed that I wasn't carting my Canon around as much as I used to with my Olympus. I started missing quite a few of the the opportunistic shots that I used to love. I had also made the mistake of retiring the Olympus gear into my "museum" in my lounge room. I started to look longingly at my (now unused) compact Zuiko lenses... Until...... Oh what the hell!! - I went out and bought myself an Olympus OM-D E-M1 together with the 12-40 1:2.8 PRO lens and an MMF-3 4/3 adaptor. My credit card statement for last month was a shocker! ;-)
The E-M1 has now become my "take everywhere" camera, with the EOS 6D relegated to roles when I know I am going to be taking photos and especially where the light might be questionable or require a great dynamic range capability. Don't get me wrong. - The EOS 6D is a more competent camera and offers superior image quality in low lighting conditions. I also find it easier to use and, just for good measure, it fits into my hands more comfortably. But for ease of carrying around, the E-M1 has it beat without much loss of functionality.
The auto-focus on my old 4/3 lenses when using the E-M1 is fine during the day, but is not particularly good under low light conditions. Using the micro 4/3 12-40 lens, it is as good as, possibly even better, than that of the EOS 6D. Under decent lighting conditions, I struggle to see a difference in image quality.
From the work point of view (I am a civil engineer with responsibility for road operations for local government), I value a camera with GPS ability for geo-tagging my photos. Both cameras have this ability. - The E-M1 syncs with my iPhone 5S and the EOS 6D has GPS built in.
My advice? -> If your reason for getting an E-M1 is to use it as a "take everywhere" camera so that you do not miss opportunistic photographic opportunities, go for it! - You won't regret it! Just don't expect its low light capabilities to be as good those on current model DSLRs.