I have the same problem OP. Flat feet cause them to pronate the wrong way when you walk which causes lots of other problems as a bi-product. I ended up going to the foot doc and getting insoles made in the exact shape of my feet (or how they are shaped if I had an arch). Very costly but it was worth it.
Another thing that helped greatly is buying shoes that have no built in arch. Most companies have shoes that are marketed specifically towards people like us. New Balance has some great ones.
I ended up having a stress fracture in my foot due to running a lot with flat feet and now I do more biking for cardio instead.
One thing you could try is wall stretches, place one foot back far and push against the wall to stretch the plantar fascia, hold this for 60 seconds, change feet, take a quick break and then find a ledge (a couple inches high, curbs work well for this) and place your toes on this and with your heel on the ground, get a good stretch in again for a count of 60. Repeat these stretches 3 times, for a total of 6 stretches for each foot, and it should have everything pretty loose and warmed up. There are other stretches you can try too, just google plantar fascia stretching. Being flat footed allows it to get tight which causes a lot of the foot pain. You could also take a supplement to reduce inflammation. Ibuprofen works but I don't like to take it because it isn't great for my workouts. You should also finish your run with some more stretches and soak your feet if you can remember.
Hope this helps.
