I've had Sony mirrorless cameras since the Nex 5 came out, have upgraded along the way and unfortunately I'm writing them off. Along the way I've had 1 camera pretty much die, 1 lens completely die, 1 lens only works once in a while and their flash system has never seemed to work as well as my Nikon system. It was for me a nice experiment and a good way to use some old Leica lenses I had sitting in a closet... but honestly their products just seem to be holding up that well. I've got a lot more Nikon stuff and have only had one camera die and it was older than the oldest Sony and was well used.
If you do decide to go for Sony anyways then look long and hard at the lenses that are available and their prices. Sony doesn't offer nearly the number of options that Nikon or Canon offer and what they do offer can be pricey.
One last thing on Sony, we have had decent experience with their RX100 camera which will probably give you results close to what you are getting with the 5D3 using the kit lens except that you wont have quite as good of lowlight performance and the zoom range tops out at 74 isntead of 105... If you wanted more zoom then a RX10 would give you more than what you've got now in terms of zoom with pretty much identical performance to the RX100.
Can't say much about Fuji as my only experience with them was a point and shoot that my wife had so it really never got much use other than as more clutter for her purse.