Hi Lisa, This is Dr Emily Weaver, I am an experienced equine vet and hope I can answer your question!
It certainly sounds like Kat may be experiencing Inflammatory Airway Disease (IAD), which is an inflammatory condition of the lower respiratory tract that is primarily non-infectious of origin. Most likely it is due to sensitivity to inhaled allergens. The most common allergens we see in the horse are dust and molds in hay, and pollens or dust in the environment.
Kat is only 4 years old, so I wonder if she may have had some type of respiratory infection in her past, that then may have predisposed her to being sensitive to allergens in her environment? Sometimes with IAD, it can predispose them to secondary respiratory infections as well, and even though she is not acting sick she may have a low grade respiratory tract infection going on that could be contributing.
The most important thing when dealing with IAD is trying to make environmental changes. It will be difficult if where you are is very dusty, I don't know if you can mist her stall to keep the environment less dusty? I would also recommend soaking her hay. You need to completely submerge it in a tub or large bucket for at least 30 minutes before feeding. Many of my clients soak the hay in a hay net so it is easy to lift up an out of the water after it's done soaking.
There are medications available that will help with the IAD, and it sounds like she may need to be put on some type of steroid (I usually use dexamethasone or prednisolone) at least temporarily while you make the environmental changes and hopefully get the cough under control. There is another drug called Ventipulmin, which acts as a bronchodilator that I also will prescribe, but it is pricier than the steroids. Depending on how she responds to environmental changes, she may need to be on long term medications, one or the other or a combination in order to get her symptoms under control. In a lot of horses, IAD is seasonal, so she may flare up in the summer, need meds, then be fine during the winter times.
I would recommend an exam by your vet, make sure they listen carefully to her lungs for increased airway sounds and wheezes. I would also recommend a CBC be pulled and look at it for signs of infection. She may need a round of antibiotics if it looks like she has a low grade infection. Your vet can then prescribe other medications for the IAD that they deem appropriate.
I hope that helps, and let me know if you have any further questions or concerns,