We also used the pop up mesh hampers. We always used the car to take the hampers to the laundromat so did not have your problem with that issue.
One suggestion is a fold up luggage cart/dolly. It folds up flat for easy storage and could easily handle a mesh hamper. I think we got ours at a Walmart.
After we bought a house in CO and had all of our stuff moved there from 11 years of storage in NH and staged in the shop, my wife used it to move boxes from the shop to the house for unpacking. It has held up fine for that duty so should work great for laundry.