I have had the same problems with both Folding and Non-folding tiers. Often the Bead is off or thin or too close to the wire.
Take your tire and inspect the tire closely measuring the thickness and width of the bead. I think you will be surprised.
If the bead checks out as good then most likely it is slipping off the rim where the fold was at. For that problem I mount the tire then use zip ties to hold the tire inside the inner rim as I inflate the tire. Then after letting remain in place for a few hours I will remove the tie.
OK... Got My Three Flats
Tire Not Seating - Another Thing To Check