I don't have any direct experience with that exact combination of gear, but it's almost certainly something to do with video levels. Whereas the file in which the video is stored is capable of recording brightness levels between (assuming 8 bit) 0 and 255, which is sometimes called "full swing", many video devices only use the 16-235 range, which is sometimes called "studio swing". This was initially done so that certain voltage levels from the analogue world would be easy to maintain when converting to and from digital video, but these days, it's pretty much just an annoyance.
The 5D Mk. 2 was always famous for recording full-range, 0-255 video in such a way as to cause lots of postproduction software to assume it was recording limited-range, 16-235 video. The result of this is that the blacks get crushed and the highlights get blown out, to a degree that's sufficiently similar to what you're showing in your stills that I assume that's what's going on. Theoretically, this should all be automatically handled, using markers in the file, but it is rather poorly standardised and your problem is common.
If you want to prove it, sometimes you can establish more firmly that this is the problem by pulling it into something like Resolve or just After Effects and pushing the black level up, at which case it is sometimes possible to recover the lost detail, depending how the software is behaving in that particular circumstance. Sometimes the lost information gets cropped off and lost, sometimes not.
One decent solution is a piece of software called 5DtoRGB by Rarevision, which will do ProRes transcoding from 5D3 files (and those from other cameras) in such a way as to handle the video levels correctly. It was developed specifically to handle this sort of material and knows how to do so correctly. Disclosure: I know the developer, but I'm not financially involved. There are probably other things that can fix it. I'm not sure what.
If you just need to supply the material to someone else you can do so in confidence that there's nothing actually wrong with it. All 5D Mk. 2s and I assume 3s, and a lot of other cameras, do the same thing - and if the question comes up you can answer it for them.