Just guessing here but to me it looks like a contact dermatitis of some sort. Check your fields for plants, bushes, or trees that could create a striated pattern (as shown on your horse) from the horse rubbing up against it or walking/running past it. Kind of like a person who gets red lines that show up after scratching themselves. My sister scratches her arm for example and the scratch line turns red and puffy. I agree that an oral steroid may help. If your vet orders it, horse takes it, and skin problem disappears..you have your answer! But again..just a guess. Good luck.
What kind of surface are your horses sleeping on? There are few lines in nature this uniform which leads me to think it is from something physical they are in contact with-grooved mats, concrete, that kind of thing.
They may also be reacting to something causing an initial reaction then the lines are secondary.
What, if anything, has changed in the past month?