A horse’s coat begins to grow when the days get shorter and not as the temperature drops. This is because a horse must begin to develop his coat long before winter sets in. As the days grow shorter, the horse’s retina registers fewer hours of daylight and the brain is prompted to release extra melatonin, a hormone that tells the hair follicles to grow hair.
If the horse is being allowed to grow his natural long, heavy winter coat and has not been clipped, you have to remember why you're letting this happen. If your horse is going to be out in a field---with a shelter, of course---for all or most of the winter, that long, healthy hair coat will keep him dry and warm. His winter coat will be oilier than his summer coat, and that's fine; the oils help him stay warm and make him more water-resistant. The only problem comes when you forget that you're dealing with a special-purpose winter coat and try to groom him as if he still had his summer coat. Don't. Removing all of the oils will defeat the purpose of letting him grow that protective coat, so when that winter coat begins to come in, you should limit your grooming accordingly.
When the winter coat is established, a little light brushing and rubbing with a cloth, such as a cactus cloth, will leave the oils in place.
If you haven’t clipped the horse, training in the winter will present a conundrum. Just how can you dry out that thick heavy coat? There are two useful methods for dealing with a wet horse in cold weather. One requires electricity, the other doesn't. If you have a horse vacuum cleaner, set it on "blow" and you can blow-dry the horse's coat. If you don't have a horse vacuum cleaner, don't despair - many inexpensive shop vacs work very well on horses, and although shop vacs don't typically come with horse grooming attachments, there's a curry-comb attachment available through most vet supply catalogues. These gadgets are brilliant. You can reverse them to "blow" and blow-dry a damp or wet horse.
The low-tech option is to put a large cooler made from wool or Polarfleece over your horse and walk him until he is cool, but not necessarily dry. Then put him on crossties in the aisle or in his stall, and let the cooler wick the rest of the moisture out of his coat. If he's still wet, you can ‘thatch’ him by taking a few large handfuls of hay or straw and stuffing them between the top of the back and the cooler. This creates a larger airspace and helps his coat dry more quickly.