I have towed it with a V8 Grand Cherokee, Lexus GX (V6), Tacoma (V6), GMC Yukon (6.0 V8), and Hyundai Palisade (V6)
all have been fine, though I haven’t crossed the continental divide. I did the run from the northeast to Florida using the Grand Cherokee and Tacoma, both equally fine.
The biggest difference is that getting weight moving is actually not that challenging. It’s mostly a function of the rolling resistance of the tires. The bigger difference is the trailer does not have brakes actuated by a controller in the vehicle. The brakes manage themselves and the tongue is quite light - the trailer has a handle on the front of it and a person can actually move it if the parking brake is not on.
It is much more like flat towing a vehicle than it is like towing a conventional horse trailer. It also has independent suspension more like a car, so the horses shifting does not directly translate to the tow vehicle.
I would not recommend hauling it without a vehicle that has some hauling preparations, but especially when using a vehicle with a modern torquey turbo engine I’ve had zero issues with it in 6 years of towing. The least pleasant towing experience was actually using an F250, which was way too large for it and it wanted to bounce around a bit behind something that large.