Personally, never, never, never kilts of any kind in the dryer. Ever.

Want a canvas / cotton / blend fabric softer(?); wear it more.

Favoured way of "cleaning," as cited by others, swab the especially offensive area with a clean cloth, cold water and mild soap while the kilt is laying flat. Rinse it the same way, multiple times.

Hang to dry. Touch-up press pleats if necessary using moist cloth and iron not touching the kilt fabric. Be careful of the iron's heat setting.

Only one wearer's opinion....

Best of fortune.