I've been doing equestrian photography professionally for over 15 years. And have guided more than a few workshops for folks wanting to learn to shoot these events.

Dealing with indoor arenas is just part of the game.

I assume by "holy trinity" you mean something like 16-35mm f/2.8, 24-70mm f/2.8 and 70-200mm f/2.8 lenses... you will need the latter two lenses. I'd put the 24-70mm on the D850 and the 70-200mm on the D750. If you have a 1.4X available, take it to use on the 70-200mm. I make a point of shooting with APS-C (DX) cameras and often use up to 300mm when shooting lengthwise in an arena. You are using full frame (FX) cameras, so you won't have the advantage of "reach" that I do with my crop cameras.

You probably won't need the wide angle lens, but I carry one "just in case", to make an occasional wide shot of the venue or perhaps a silly shot of a horse from close up.

I always use at least two cameras and do much of my shooting with 70-200mm (112-320mm equiv.) on one and with a 300mm lens (480mm equiv.) on a second camera. I have 300mm f/4 that I can use handheld when the lighting is sufficient... but if needed I have 300mm f/2.8. It weighs 6 lb., so isn't a lens I can hand hold for more than a few minutes. So it usually goes on a tripod (with a gimbal) or at least a monopod (which leaves me more mobile). When able to do closer shots I swap the 300mm to my 24-70mm (28-112mm equiv.). In some of the "worst" lit arenas I occasionally have to resort to faster prime lenses.... 28mm f/1.8, 50mm f/1.4, 85mm f1.8 and 135mm f/2. (Note: Outdoors and in better lighting conditions, instead of the 300mm I use a 100-400mm f/4.5-5.6 lens. But it's not bright enough for enclosed arenas, but is versatile when I can use it.)

With my current cameras I'm able to use ISO 3200 without much concern... 6400 and even higher with some extra effort in post processing. I know I need a minimum of 1/500 shutter speed to have any hope of stopping action... Even that's a bit too slow though. 1/640 or 1/800 is more what I try for. At times I've used as high as ISO 12800 and 16000! Even if it's a bit noisy, I figure it's better to "get the shot" than not get it at all! I will sometimes convert to black & white if noise is an issue.

Often arenas will have strongly backlit areas. Watch out for those. If you use an auto exposure mode, it will want to underexpose and when you try to correct it in post you'll greatly amplify any noise in the image.

Also watch out for mixed lighting conditions. This can make for odd color casts. In some arenas I have to set a custom white balance to avoid ugly tints from the lighting. One arena I work a lot, I actually ask them if possible to turn off the lights (but still get some mixed lighting because of skylights in the roof that have yellowed with age). If the lights are on there, a horse can have a magenta tint on his face and a green tint on his rump! Makes for really difficult corrections in post processing.

I always shoot RAW to have the most flexibility to correct things in post processing. If for some reason I want JPEGs, I'll shoot RAW + JPEG. That fills up memory cards fast, though. Especially when you take several thousand images at an event!

You have to be a bit careful taking photos around horses. I have no problem with my APS-C cameras noise levels... even shooting bursts at 10 frames/sec. But my full frame camera's shutter noise is loud enough to cause an occasional horse to "go vertical"... not something you want to do! When in doubt, I try to carefully pre-test a horse's reaction to the shutter noise. I have used flash with horses... they tend to completely ignore it. In fact, often riders are more bothered by the flash, than their horses are! But flash doesn't work when shooting bursts of images, so I very rarely use it.

Besides the noises of your cameras, you also need to be careful not to startle horses with too quick movements or loud noises. Don't "hide" behind anything or crouch down like a predator about to pounce! Stand where the horse can see and identify you as a human being. Don't walk up behind horses... They can kick you to oblivion! Especially watch for horses that have a red ribbon tied in their tail... that's a universal warning of a horse that's prone to kicking.

Most horses are very good natured. Some even seem to have a sense of humor, like the one I caught chewing on my 24-70mm lens or another that wouldn't stop "grinning" at me (see the "portraits" below)!

I use continuous frame rate and shoot short bursts because some things are hard to time, such as the gait of a horse. I'm trying to get the horse and rider in favorable pose, so am timing for peak action such as going over a jump or around an obstacle or making a particular movement in dressage. But movement of the horse's legs is even faster and there are some positions where it just looks awkward, frozen in an image. There are also certain movements that are near impossible to photograph in a complementary way... a dressage move called a pirouette is one example that comes to mind. A photo just can't capture it well.

I use continuous focus too (AF-C in Nikon terms)... with back button focusing. I often hear other people using One Shot (Canon) or AF-S (Nikon) at events... and can guarantee that a lot of their shots will miss focus when the subjects are moving!

And with my DSLRs I almost always use a single AF point, keeping it right where I want the camera and lens to focus. Usually I put the AF point on the horse's chest since that will also put the rider's face in focus... but unless I can stop down a bit, when the subjects are closer the horse's face can be out of focus... so at closer distances I may focus on the horse's ears instead.

You didn't specify what type of horse show you'll be shooting... and that can make a difference.

- English-Western shows tend to have slower movements... a lot at a walk or trot and even some stationary posing.
- Dressage is highly organized and specific steps that are ridden in predetermined patterns. Usually they start with simpler patterns with fewer steps, then get progressively more and more complex and advanced. There also may be events with multiple horses and riders (Pas de deux), and some are ridden to music.
- Hunter-Jumper shows are generally quite active... things happen fast, but each competitor in a class will take the same route (or is supposed to), so you can have an idea where they will be going.
- Gymkhana and rodeo have a lot of high speed action around obstacles or chasing cattle or whatever. It can be pretty unpredictable, too.
- Trail Trials don't happen in arenas and cover some distance, with 6 to 10 obstacles the horses and riders need to negotiate along the way. Since I have to hike up to 10 kilometers at these events, I lighten up on the gear I carry.

Some of my favorite equestrian shows have been miniature horses and miniature donkeys! Particularly with the mini donkeys, the mistakes and "blow ups" are every bit as much appreciated as a "clean run". When everything goes wrong (sometimes with hunter-jumpers too) I'm often asked, "Did you get that? I want a print!" Other shows are a lot more "serious".

You're welcome to check out some of my stuff over on Flickr. I only have a small selection there... and some of it is there to illustrate what not to do or what doesn't work... but the following may give you some idea what to expect:

English-Western: https://www.flickr.com/photos/amfoto1/albums/72157626378755889
Dressage: https://www.flickr.com/photos/amfoto1/albums/72157627539635849
Hunter-Jumper: https://www.flickr.com/photos/amfoto1/albums/72157619801710947
Gymkhana: https://www.flickr.com/photos/amfoto1/albums/72157626503423304
Trail Trials: https://www.flickr.com/photos/amfoto1/albums/72157627847319536
Portraits (not all equestrian): https://www.flickr.com/photos/amfoto1/albums/72157631128669784

I "work the rails" of the arenas I shoot, but some may limit access in certain areas, to avoid problems. For events that are judged, I'll often set up near the judge because they have a good view of things. But don't be afraid to move around and try different angles, as best as you can safely do so. In some classes I'm actually in the arena with the participants, but you probably won't have that kind of access and probably shouldn't go there anyway, unless you know horses and the events well.

Finally, keep your head on a swivel! There are often photo ops among the spectators, interactions between people and horses or between horses with other horses. Also you need to be aware of horses around you for your own safety.

I have the D850 and found it to be very good using high ISO! I would probably use the 70~200, but wouldn't be afraid to use something longer if you have something that is fairly fast. I wouldn't be afraid to use an f4 if it's a tried and true lens. Personally I would use my 200~500 if necessary for longer distance work. A monopod might be useful, if allowed. Otherwise I would look for some structure in the stands to brace against. But mainly I would have fun! JMHO.