From the manufacturer:
Walnut oil is a natural oil that does not evaporate or remove essential oils from the artist brushes. The addition of walnut oil to color will slow drying, enhance flow, and increase sheen. Our walnut oil is designed to augment the special nature of our oil color but it is completely compatible with other artists oil colors and mediums.
Walnut oil has a unique refractive index. Colors ground in this fine oil are more jewel-like and bright. Using walnut oil allows us to increase the amount of pigment of each color resulting in paints with the highest possible mass tone and tint strength and thus creating paintings saturated with extraordinary color.
Walnut oil does not yellow as much as other oils. This is especially important in whites, blues, and yellows. It does dry a bit slower than linseed oil, based on environmental factors and paint film thickness, about 20% slower.
Most artists find that they don’t need to thin our paint but if, for your particular technique, you want the paint more fluid or creamy then add a small amount of walnut oil as in 1 – 3 drops per inch of extruded paint and mix it in. You can always add more but do it in small amounts until you get the desired consistency. There is not an exact measurement or rule as every artist and technique is different but we do caution about adding too much as your paint can become too thin and oily. The addition of walnut oil can slow the drying process.
It is important to realize that the masters did not have petroleum based solvents. Because of that they had to clean their brushes with, in most cases, walnut oil. And here’s how you can too! Wipe the excess oil paint off your brush with a rag or paper towel. Dip your brush into a container of walnut oil. We like to recommend the Silicoil tank with armature wire. Then using your rag or paper towel wipe the oil paint off the brush. Repeat this process until the paint is removed. Be sure to wash your brushes with warm water and a mild soap like Dawn, Pink Soap, Murphys Oil Soap, or any brush soap at the end of the day. If you leave oil in the brush it will dry and then the brush is unusable. Continue to use your walnut oil over and over again until it will no longer slide the paint off the brush. The main difference between walnut oil and solvents is a solvent dissolves the paint; walnut oil makes the paint slide off the bristles of the brush. Cleaning your brushes with walnut oil is safer for you and the environment as well as its easier on your brushes.
For oiling out we recommend using walnut oil instead walnut alkyd. After the oil paint has cured (dried), take a soft cloth and apply a very thin layer of walnut oil to the surface wiping 90-95% of the oil off. You want to leave a very thin even layer.