As encaustic medium does not deteriorate the brush and can always be remelted, brushes can be left uncleaned indefinitely.
No solvent is necessary for cleaning encaustic equipment. Should you wish to clean your brushes, you can dip them in melted paraffin or soy wax. Soy wax is non-toxic and burns cleaner than paraffin. Soybeans are a renewable source, unlike paraffin. Soy wax is naturally biodegradable. Soy wax is also easier to remove than paraffin wax, so after the colour has been cleaned out of the brush, the brush can be washed with soap and water and is reusable in other mediums.
Enkaustikos manufactures a Slick Wax specifically for cleaning your encaustic brushes, tools, and even your palette between encaustic color changes. Rinse your brush in the slick wax and wipe off any excess wax with a paper towel, the slick wax will have removed the color.
Encaustic tools can be cleaned of wax and paint by leaving them on a hot surface and wiping them clean when the wax has melted.