Sarcasm aside, I wouldn't say anyone shoots birds at "night." I've shot some Night Herons "at night"...as in it was quite dark, late twilight, not post-sunset but "night":
Not in flight, though...that's pretty much a miserable failure at night regardless of the gear.
Birds are often quite active during crepuscular hours, where the light can be many orders of magnitude lower than during full daytime. Some of the most dramatic BIF shots I've seen come from these hours...so having a fast lens and good AF system that can handle it (if BIF is a primary goal...that's what we are talking about here, BIF as a primary form of photography, not something done occasionally) is very important.
In great light, just about any camera will do, and just about any f/5.6 lens will do as well. However, if you don't want to be fighting against the gear in non-ideal circumstances, I think you have to be realistic about what kind of equipment you
need, vs. what kind of equipment you
can get away with. I think you can get away with a 6D and 100-400mm, for sure. Even the 100-400mm Mk I. I cannot recommend that combination if you want to actually really DO bird in flight photography as a primary thing. If BIF is a primary form of photography for someone, I feel it is my responsibility to steer them away from the "get away with it" gear, and towards the "nail it every time" gear.
In all seriousness.

No sarcasm here. (I guess I woke up on the right side of the bed...oh, right....I didn't wake up at all, guess that's one benefit of insomnia, no side of the bed to worry about, period! ???