There are a lot of reasons that could cause an improper kicking when it comes to swimming breast-stroke.
I had a few students with flat foot before. They can’t squat with both feet firmly on the ground and usually squat ‘tip toeing’. (if you understand what I mean). Flat footers cannot balance themselves well if they try to squat. Other than that, for normal leisure breaststroke, they can kick quite normally. I was not even aware they have flat foot when they are swimming till they are doing the squating dive.
Thus, this lead me to believe that flat foot does not really affect breast stroke that much unless you are looking for them to go into competitive swimming in which every little details will count for the extra speed.
Doctors have also recommend swimming as one of the exercises for flat footers. When flat footers do long distance jogging, they will sometimes feel pain in their feet and back, and swimming takes the load off the feet and thus does not strain the feet that much.
A general problem that people have with breaststroke kicking is that they don’t dorsi-flex their feet. (I am guessing this is the problem with your son but have to await your confirmation) Simply put it, dorsi flexion is when you pull back your feet towards your leg and your feet form a L shape with your leg. When you dorsi flex and kick out, you provide a bigger surface area with your feet to push the water back and move you forward.
On the other hand, when you do the front crawl/freestyle, your feet need to be plantar flex. Plantar flexion means that your toes are pointed away from the leg and form a straight line with your leg. It is like when you are tiptoeing.
Some children have this problem of being able to kick nicely for front crawl but have form a habit so when they do breaststroke, they will kick out with the feet plantar flexed instead of dorsi flexed. This will cause them to move quite slowly.