My DSLR is still my primary BIF tool, but I replaced my NEX-5N with an A6000 hoping it would finally open up being able to use AF-C and tracking modes for some regular BIF use. And indeed, the combo has lived up to the promise - it's shockingly good at BIF, keeping up with no problem, even with smaller, faster birds. The limitation is as mentioned, the lenses - the longest e-mount lens I can get is a 210mm F6.3, or the 200mm F4 that was used in the test. I go with the 55-210mm F6.3 lens for one primary reason - I can attach a 1.7x teleextender to the end of it which gets me out to 357mm optically, for a more decent 535mm equivalent reach. With that combo, it's afocal in design so no light loss from the extender, and autofocus and stabilization in the lens still work exactly the same...and it stays small and lightweight at under 3Lbs total, which is less than 1/2 of my DSLR birding rig, and a good foot shorter!
Some BIF samples from the A6000 I shot this past weekend - I'm a bit more limited in bird selection going into summer, as we've lost our winter visitors and raptor activity is a bit low, so it's mostly egrets, herons, ibis, and the occasional martins and grackles...but these were all shot in AF-C continuous focus mode, set to 6fps burst, using wide focus area (the camera prioritizes the subject closest to the camera, so it even picks up birds against cluttered backgrounds, which was not always the case with the CDAF of the NEX-5N), in shutter priority with auto ISO set to 100-6400 range...
I mean these to just exemplify what the new class of mirrorless cams can do with regards to continuous focus and tracking...they're quite surprising and a definite big step up from previous mirrorless. For BIF shooters, they're a match for pretty much any entry-DSLR...and in some ways maybe even a touch better for the focus system itself - in ability to cover more of the frame with cross-type PDAF sensors or specialized CDAF systems in the case of the Panasonic, and in ability to track focus with very fast movement even straight at the camera while firing continuous bursts...in my camera's case, up to 11fps. The lenses are still the area that's lacking - and DSLRs still have that advantage - I can use myriad lenses that are both much longer reach and much faster apertures for my DSLR that just aren't available on pretty much any of the mirrorless cams. And while most of these can use adapters, some that even autofocus quickly, with some of their sister mounts, these would precluse use of these advanced focus systems which in some ways would defeat the purpose.
What it has done is made my mirrorless system a much more worthy and equal replacement for my DSLR - I can bring it along as a second body to cover a shorter focal range when I'm shooting a long prime on the DSLR, and still be able to shoot action and motion, or I can bring it in place of the DSLR when it's hot and sticky out and I just don't want to lug the weight around, and know I can still use AF-C mode and still get continuous focus tracking on fast moving subjects. While I shot BIFs on my NEX-5N previously, it was only the occasional shot, using AF-S, and cycling the focus myself...now with the new gen cameras, I just switch over to AF-C, wide focus, or center focus with lock-on AF mode, and let the focus system track while I keep it in frame, just like the DSLR.
Sony DSLR-A68 / Sony 18-250mm / Minolta 50mm F1.7 / Tamron 150-600mm / Minolta 300mm F4 APO
Sony A6300 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / FE70-200mm F4 G OSS / FE70-300mm F4.5-5.6 G OSS / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses