I'm not a crossfitter, but I do compete in a variety of strength sports and the following are the footwear choices I have made (my gym bag is a large duffel bag):
Because I am flat-footed as well, I rip out any insoles to prevent me twisting my ankle from having artificial arch-support (arches cause my ankles to roll outward).
Addidas baseball cleats for Highland Games
Custom Olympic Weightlifting Shoes for "most" gym lifting (squats, cleans, standing press, pulls)
DoWin Olympic weightlifting shoes for strongman stuff (I baby my custom shoes and won't use them "off platform") such as stone lifting and overhead stuff.
Vibrams for some outdoor strongman stuff, box jumps, etc... (I also do the majority of my walking and running in Vibrams). When I deadlift instead of doing clean pulls, I either go barefoot or just wear the vffs.
Old-school Vans Off The Wall hightops for any "walk with weight" events in strongman like Yoke, Farmers, etc...
Since I don't really compete in strongman anymore I haven't had the need to get rock climbing shoes for the truck pull, but as I was a rock climber in high school it might be something else I add to my bag in the future.