The first assumption to answer that question is the assumption that there are running shoes that are helpful for flat feet (‘overpronation‘) and those running shoes actually do what is claimed and also that something needs to be done about the flat feet. This issue is not without controversy with differing opinions and mixed results from the research.
Secondly, we are assuming that something needs to be done about a flat foot and that is not always clear cut and obvious.
Thirdly, there are multiple causes of a flat foot, so anyone giving a blanket recommendation about what to do with a flat foot (or ‘overpronation‘), then avoid their advice.
– if your flat foot is due to tight calf muscles, then use lots of stretching and a running shoe with a high drop (high difference between the height of the heel and forefoot).
– if your flat foot is due to a problem with control around the hip joint, then the shoes are probably not going to make a lot of difference and you need to work on the hips and running technique
– if your flat foot is due to a more medial location of the subtalar joint axis, then you will probably benefit from running shoes that have medial posting in the midsole
– if your flat foot is due to a bony forefoot varus, then the only thing that can help are foot orthotics and the choice of shoe will probably make no difference (and neither will muscle strengthening or changing the running technique make any difference here either for this cause).
– etc etc …. I think you get the point. This list could go on for quite a bit.
If you want to get a running shoe for your flat feet, then get some good advice about what is causing your flat foot first and then see if there are any design features in the running shoes that can help your particular cause. Above all, don’t listen to an avoid most of the nonsense on this topic.