My trailer’s interior lights come on automatically when the headlights/running lights are on. I can turn off the light in the tack room but the horse compartment ones stay on. They are a soft glow so it gives just enough light but doesn’t blind them like some of the bright LEDs could do. FWIW my trailer is older though, an early 90’s model.
I travel with the interior lights off. My light has a small switch on it to turn it off. To those of you who think your lights come on automatically and you don’t have an option, get up on a bucket and have a closer look at your lights and see if they actually have a switch. I’m not trying to make anyone sound stupid for not realizing they have a switch, but I have shown at least two Pony Club moms where their light switches were because they didn’t think they had them.
Shrug. I’ve left the horse box interior lights on accidentally before. It’s not always intentional.
By accident I found that having the lights on while traveling at night keeps my horse quieter. I think headlights blasting thru the windows would kind of spook him. He doesn’t stomp around (as much!) and his hay bag is emptied more often. I turn the loading lights on before I open the rear doors so he sees the ground as soon as I turn him around. It works for us. 
