Protecting from the soiling of the body is easier since undergarments made from linen or wool along with other layers protected the "fancier" outer garments. Plain pieces that could be bleached, scrubbed, boiled, and washed as much as need be would absorb the sweat and dirt. Basic washing of the body daily along with changing of these undergarments kept people fairly clean, though "bathing" was not as common (most of us don't bathe, but shower today).

For the finer pieces dealing with outside dirt and stains there were many ways to clean. A good example would be a grease stain, possibly from a wheel or even food. Water won't get it out alone and using lye soap can be very harsh. Using fullers earth, a powdered dry clay, to soak up the grease is commonly recommended. Sprinkle it on the area, cover in paper, and use a dry hot iron to draw the grease out. Then gin could be used as a solvent for the area and water to pull the gin out. Instructions on cleaning methods, both dry and wet, can often be found in the back of cook books. Surprisingly enough though, most dirt and mud brushes off easily once dry.

Keep in mind that a great deal of extra pieces and detachable items in the clothing can make cleaning easier as well. The jewels around the neckline would be loosely sewn on and detached from the kirtle easily for cleaning. Facings in the hems of gowns with trains would be meant for replacement, protecting the finer fabrics from the ground.

It doesn't quite go back as far as the Tudor period, but check out the book Hubbub by Emily Cockane or delve into some of the Tudor Tailors research and books on that specific era of dress. The Dirt on Clean is on my reading list for the topic as well.

In "Between Pit and Pedestal, Women in the Middle Ages" by Marty Williams and Anne Echols, clothing could be stored with herbs like lavender so they would smell nice. Surcoats could be worn over gowns and sleeves could be knotted or tied up to avoid dragging on the ground or in food. Aprons could also be worn to protect the clothing. The rich/noble could afford new clothing when it was soiled, or employee someone solely to clean clothing.

Also, in "A Distant Mirror" by Barbara Tuchman, people in the 14th century could wash some clothes like we do now...in a tub and hang to dry. Nobility and other upper class people would have places to put clothes, like a trunk or closet. They could also remake their clothes when fabric was ruined or dirty. They could brush out velvet or beat the dirt out of clothing like you might a rug. If you were poor, you pretty much had one set of clothing anyway, and it would get pretty dirty. They didn't bathe that often either.

So while this is a few hundred years before the Tudors, I think it would still make sense.