Had one go on my old PuP once. They were due for replacing but I hadn't done it (bad me). The trailer started pulling to the right, and made a loud thump-thump sound after the air got low enough - it didn't pop, it just deflated over a period of a half a minute or so. I pulled over and changed it. No damage.
I also had one go on my current trailer (caused by the piece of metal that was still in the tire) but it's a tandem axle. It was making a similar noise, but there was no noticeable change in handling. I suspect I was traveling a lot further with it in bad shape. No damage there either, although changing it in the rain was no fun.
2021 Four Winds 26B on Chevy 4500