While you can harvest actual dirt and dust from your local area, baseball fields, and deserts, it’s often better to purchase dust that is fine and free of pebbles and impurties. There are many dust products on the market. The classic movie dust is fullers earth, a tan-colored fine dust. Alternatives include diatomaceous earth and walnut dust.
One of the classic methods for dusting items is to use a flour sifter. You can apply it directly using a rag, or a tied-off piece of panty hose or a stocking. The stocking allows plenty of dust to fall out with ever pat. If you want really cover the garment in a layer dust, or give the appearance of thin, dried-out mud, dilute the dust in water and apply it or spray it onto the garment. All you need is a couple tablespoons in a large spray bottle.
For a more permanent effect, seal it in with a clear spray, like a matte spray paint, or a sealing spray. This will be much more important for plastic, since the dust will come off of it easier than porous materials.
One common dust that you can use is talcum powder. It’s lighter colored than fullers earth, but when combined with other weathering effects, can add nice dusty highlights to the item. Flour and corn starch should be avoided since they are prone to decompose.
A permanent way to add a dusty or thin mud effect is to use matte paint. Application, however, may be challenging. Your best bet is probably to water down the paint, and then to spray it, or apply it with a rag or sea sponge. Be careful with spray paint, as most of them contain solvents that are not compatible with vinyl. Use a vinyl spray paint instead.