Could be many things, but few things i find when riding road, firstly my feet swell after a while, so often I have to loosen my road shoes a little after about 30 minutes to what have felt a little slack when i first started.
Secondly, make sure when you are riding flats that pedal is in right place, with SPD’s the axle of the pedal is supposed to be in line with the widest part of the foot, with flats your foot should be more central on the pedal so the pedal axle is more in line with the arch.
Also as above, make sure the arch of your foot is supported correctly with a decent insole, the cheap Speciallized ones are pretty good and they can fit them correctly in those LBS that sell them.