Then why has no one yet suggested using a litter?
First, get a dirigible …
If the mountain will not come to Muhammad, then Muhammad must go to the mountain. Instead of taking litter to the cat, you could put up a ladder so that the cat can do its business elsewhere.
Stairs?
Well, if you’re using a trebuchet, you don’t need windows that can be opened…
You just need a way to replace the glass after the delivery is done. 
For that matter, a catapult, canon, or giant slingshot would serve in the same manner. 
Seriously, though, do you think these ladies can haul around the materials needed to build a siege engine if they’re too weak to haul around 20-pound containers of kitty litter?
C’mon, guys, the stuff is fungible. If there’s an open-able window or door, I’m thinking* it’s better to break that container up into millions of tiny subsets:
1 Portable shop-type blower/vaccuum. Walmart sells some that weigh about 10 pounds or less.
40 feet of extension hose (you’d have to order from the shop vac manufacturer/distributor)
The original hose for the vacuum
1 or more spare pillow cases.
Start at the destination.
Connect the extension hose to the suction side and hang the other end down the stairwell or over a balcony.
Connect the original hose to the blower end and stick the other end of it into a pillow case.
The lady upstairs controls the switch and gets to change pillow cases when one is full.
The lady downstairs directs the end of the pick-up hose to keep it sucking up litter until there’s no more left.
–G!
*Yeah, yeah, I know: there’s a whole lot of fluid dynamics and ducting calculations I’m missing here, particularly for a 3-storey vertical lift and what will probably be corrugated/flexible hose. I already admitted in my Ducting thread that I’m not qualified to do those calculations. Ultimately, this suggestion is probably about as viable as the trebuchet, catapult, canon, and slingshot ideas.
I concur. As someone who hauls heavy stuff up stairs with a hand truck on a regular basis, I can tell you it’s a struggle to pull those things up stairs.
The obvious answer is that the OP (who it sounds like lives on the first floor) where the 20 lb. container of litter is, and her friend on the 3rd floor, should just trade apartments. Bring the cat to the litter.
Yeah, all these crazy ideas like asking somebody to help (which would be the best option to me) or splitting up the load into smaller loads (the best option not requiring other human beings). Come on, think about this: if they can’t handle 20 lbs, they certainly can’t handle a hand truck. That’s the equivalent of a bit over two one-gallon jugs of milk. If they’re having difficulty moving that weight, a hand truck is not going to help, given its weight and the strength required to move it up the stairs. So I’ll sit here and “laugh” at your crazy idea.
I say just Shawshank the problem. Fill your pockets with litter every time you climb the stairs, then empty as you pass by the litterbox. For more efficiency, you can fill your pockets with Kitty Roca on the way down.