For small stains or marks I would just spot clean with a soft cloth, warm water and a very small amount of little gentle soap.
If I need to wash the whole carrier for spit up or something similar I put my carrier in a mesh bag. You could also use a pillowcase closed at the top with a rubber band or hairband.
I have a front loader so I do not always wash my carriers in a mesh bag although I do always wash carriers on their own. If you have a top loader with an agitator I would always use a mesh bag or pillowcase to prevent any part of the carrier from getting caught in the agitator. I also buckle all my buckles on a soft structured carrier and release the webbing keepers.
I put just a small amount of detergent in on a delicate or handwash cycle which is a cold wash and cold rinse. If you are washing a carrier with bright colors or fabrics I would recommend a color catcher to stop any color bleeding especially during the first wash. For wraps with a linen or hemp content it is important that you use a liquid detergent without optical brightners. Powder detergents can get into the linen or hemp fibers and cause damage to your wrap. I wash all wool wraps by hand with wool wash.
After the wash cycle I hang my carrier to dry on a rack. You may fluff in the dryer with low or no heat after the carrier is dry to soften it up if needed.