For my smaller screens I use a Stanley rolling tool case. I got it at Home Depot for $59.99 (actually I have 4 of them). I don't see it on Home Depot's website, but I have seen them in most Home Depot stores. It is online here for $85 plus shipping, but I wouldn't pay this much:
This is what it looks like (with a Sima 12' screen in it):
This might fit your 7x10. Also, depending on how tightly you roll the screens, you might
be able to fit the 16x9. But, I wouldn't count on it. I just retired my 12' Sima, and I don't think I can fit my smaller Chinese screen in it. However, I started using these more for other equipment. One case holds an EON 15 perfectly. One case can also hold 2 EON 10's. I am looking into replacing the wheels with bigger inflatable ones. This case doesn't always roll easily over grass.
I've heard of people using the big trash cans also. I think it's a hit and miss whether your screen will fit or not. I guess if you could take careful measurements of a rolled screen you could measure in the store before buying. I never thought of someone accidentally using it as a trash can. Can you imagine trying to get sticky Coke off a screen?
I usually set up my screens on tarps, so I just started rolling the screen up with the tarp. This helps to protect it when moving around. I strap the rolled up screen to a hand truck now instead of using a case. I'd be interested if you found somthing bigger.
Another idea is a hand truck used for moving the big bounce houses:
I found this for about $100.