Your chestnut or bay horse can turn the color of oatmeal, with no sheen to the coat. If you do have to clip at this time, you can try an oil rinse afterwards. Add a capful of mineral oil or baby oil to a gallon of water, and rub the coat regularly. You can enhance rubbing with external applications to improve the skin and coat. Topical applications help replace the skin's natural oils and remove sweat.
Shampoos can affect the coat's shine by removing sebum.
The result can be a dry skin, which can be more prone to dermatitis. Look for a shampoo that's strong, yet gentle. It should have the strength to cleanse, but won't harm the skin's natural moisture. Be sure to rinse all traces of shampoo from the coat so you aren't dulling the shine you want. You can condition the coat with a range of products.