Here is my solution to dirty ducklings.
I use large Rubbermaid or plastic tubs and I even use a large plastic-bottom guinea pig/rabbit cage to brood my ducklings.
After going through what everyone has already described as a cesspool of filth, I came up with this (and my best so far) solution to dirty ducks.
I bought a fluorescent lighting panel insert at Home Depot which is made of hard plastic and has a smooth surface. It is a grate style and the openings are about 1/2 x 1/2 inch.
I cut this to fit the bottom of the container that I'm using and make it just large enough to be slightly elevated off of the floor. This is usually very easy to do as most containers are slightly tapered at the bottom and you will get an automatic lift off of the floor. I use a hack saw to cut the grate and it is very easy to do. You could put some type of supports underneath if you feel the need, but I've never had to do so. They are approximately 2 feet by 4 feet when you buy them and run about $10. They hold up very well.
The ducklings are on top of the grate and all the mess of food, water and poo falls right through to the below area.
You still have to clean the brooder often, but the ducks stay clean and out of the mess. The grate is easy to hose off or rinse and you don't have to dry it as the water doesn't stay on the plastic.
This has made my life so much easier not to mention keeping the ducks cleaner and healthier.
I forgot to mention that even the tiniest of feet cannot go through the openings and my Silky ducklings prove it.