I have flat feet and also wear Asics for running. I wear Ryka's for step/hi-lo. I now use cutsom orthotics but before I got them I used SuperFeet inserts...check them out!
I have flat wide feet and found Rykas to be useless. I've found that New Balance is the comfiest on my feet. I have the 608s that I use for everything.
For running, I use the GT-2110's. I've never had a problem with those. I was amazed at how my joints stopped hurting when I switched to the Asics. I'm hoping to find some good cross train type shoes for doing my Cathe's.