Lets talk the practicalities of the situation for a moment. We are dealing with an large amount of horseflesh here. Custer took about 700 mounted men to the Little Big Horn, Grierson had about 1500 horses on his raid into Mississippi during the Vicksburg Campaign, the great maneuvers in Louisiana and Texas just before WWII involved two cavalry divisions, at least 30,000 horses and mules. It is just not practical to shelter that number of animals. Even if a mounted force occupied a town there would not be enough barn and stable space to house even a fraction of the horses. In addition, using existing barns meant that force would be dispersed over a fairly wide area making command and control a real problem. Grain and forage for a large mounted force was itself a big problem let alone trying to get it under roofs.
As others have said, the universal system was to picket the animals to a picket rope strung to available objects or strung between posts carried for the purpose. The effect is like tying more horses that you ever saw in one place to a long fence. Picketing was generally done by company. When you consider that a cavalry company would have from 100 to 75 horses and that a field artillery battery had as many as 120 horses you can see that company picket lines are about as big a congregation as you would want to deal with at one time.
I had an acquaintance who as a young man had participated in the great maneuver of (I think) 1940 as a member of a “ wooden wheeled, horse drawn, 75 mm” field artillery battery. He said that the battery would meet a railroad train a mid-day, unload hay and oats and water, feed and water the animals and then move on. His claim was that in the 90 days of the maneuver the two divisions surveyed only 600 horses, that is, only 600 horses of the 30,000 or so involved became unfit for service. This was a mater of pride for him and a measure of the horsemanship and horse management of the Old Army. He said it was the biggest thing he ever did as a soldier. This from a man who went ashore in France on D+2 and fought all the way into central Germany. I do wish I’d talk to him more about the old horse drawn army before his death.