A few years ago a friend and I devised an old utility trailer as a compost (and other stuff) sifter. He always has several compost piles that are enormous. We got a old broken down utility trailer about 12' long and 7' wide that has an expanded metal floor. We use a front end loader on the tractor and put a pile of compost on the trailer. Then we tow the trailer over the garden areas and the vibrations shake the compost through the holes in the flooring.
I know this next part sounds crazy but it works . . . put a pile of compost on the trailer, drive the trailer into a garage with a concrete floor, attach an electric motor with an off-center pulley on the shaft to the trailer frame. Turn the motor on and go have a Coke, in fifteen minutes you have a nice pile of sifted compost on the floor, with a tarp on the floor, ready to be shoveled into a cart or wheel barrow. Works great.
You could do something like this only on a much smaller scale. Maybe on a kid's wagon with 1/2" hardware cloth as a bottom. When you're finished with it as a sifter you could toss a piece of sheetmetal in the bottom and use the wagon as, well, a wagon. The hardware cloth would not have to be welded, screwed, or glued to the wagon, just set it in place.
That's more or less what I was thinking -- some sort of floating/hanging tray with hardware cloth on the bottom and some sort of off-center/eccentric cam that is belt driven by an old electric motor to vibrate things for you.. That way I could just shovel until the wheel barrow beneath is full and then shut it off until it's time for the next round.. Sounds kinda fun too! If I end up making one I'll post pics..
P.S. Thanks for the info on the shopping cart sifter.. pretty cool!