I'll politely disagree on the comment that if IQ is paramount, the decision must be DSLR. Mirrorless cameras come in many different forms, and not all forms mean smaller sensors or less capable IQ. As with DSLRs, there are both APS-C and full-frame mirrorless bodies out there, as well as the M4:3 and 1" bodies...so given equal photographer skill, equal lens, and equal conditions, a mirrorless camera can match a DSLR with the same sensor size and resolution.
As for focus performance, indeed some mirrorless cameras have begun to catch up to DSLRs - not all, so you do have to be careful which model you buy. But for ability to focus quickly and with the ability to continuously focus on a moving subject, there are mirrorless camera bodies that will perform at the same capability levels as good mid-range DSLRs. Some entry-level DSLRs are not as capable as some mirrorless models, while some full-frame pro-body DSLRs are still at the top of the heap for tracking and predictive focus.
Lens availability is often the area in which mirrorless cameras lag a bit - for fairly obvious reasons since DSLRs and their lens lines have been around much longer, and because of the professional market they've long-served, they tend to have longer zooms and primes available. Not to say you can't find very good lenses in some of the mirrorless models - but generally not as many, and not covering as many different focal lengths. Of course, M4:3 and APS-C mirrorless bodies can take advantage of the crop factors of their smaller sensors, just as APS-C DSLRs can - to give an 'equivalent' framing of a longer lens.
I've been shooting birds and wildlife with DSLRs for about 8 years, mirrorless bodies for about 6 years, and overall for about 14 years. When I first started shooting mirrorless bodies alongside my DSLRs, it was to have a light second body along with shorter focal lengths so I could quickly switch between bodies for different types of shots. The early mirrorless APS-C bodies I had could match the IQ of DSLRs, but not the focus speed, and following a bird in flight was pointless to even try. I also only had a maximum 200mm lens available for that kit. I've gone through 5 DSLR models over that time, each one an incremental improvement in some area or another, and have shot primarily with a mix of dedicated primes (300mm F4, 400mm F4.5) and long zooms (200-500mm, 150-600mm). I've also gone through 4 mirrorless models over slightly less time - but each model was a big leap over the last in capability, speed, IQ, handling, etc. It's been very impressive how much things like focus speed and tracking/continuous focus have improved in APS-C mirrorless bodies in such a short time. I've also been able to add better glass to mirrorless - first with a nicer high-grade 70-200mm F4, then a 70-300mm with nicer reach. The mirrorless has improved so much that it actually shoots birds in flight better, more accurately, with higher hit rate, and just with more ease and pleasure, than my DSLR bodies. I'm also able to use a simple adapter ring to mount my Tamron 150-600mm to the mirrorless body...it can focus as fast in single focus mode, but can't quite continuously focus as well as the DSLR with the adapted lens.
Now, my DSLR tends to be used more in winter and migration seasons almost exclusively with the 150-600mm lens or 400mm prime, for birds more far away...while the mirrorless system has become the year-round primary system, best in summer and heat when the lighter weight and size is more convenient, and also is the first choice for bird-in-flight work because it's just so fast and capable in AF-C - acquiring the target more quickly, focusing with it continuously at up to 11fps, and being lighter and more agile. I'm currently using the Sony A6300, which has a 24MP APS-C sensor, 11fps continuous focusing frame rate, big buffer, lots of custom buttons, and primarily pairing it with the FE70-300mm G OSS lens, which is light, stabilized, and delivers very nice IQ. Take a look at the newer models of mirrorless for those that have improved their focus ability to the level of DSLRs - among them the Sony A6300 and A6500 (APS-C models), Fuji X-T2 (APS-C model), Sony A7RII (Full frame sensor), and Olympus E-M1 Mk II (M4:3 sensor).
Here's a running gallery I keep of all shots I've taken of birds/wildlife out in my local wetlands with the A6300 and FE70-300mm, FE70-200mm, and Tamron 150-600mm lenses:
http://www.pbase.com/zackiedawg/a6300_wetlands&page=all
I do lots of BIF shooting with this combo. With electronic first curtain shutter engaged, the shutter sound is pretty quiet, and there's an option to shoot with full electronic silent shutter for complete silence if needed.