Do they happen? I.e punter and escort becoming an item?
Of course, but I think that in general you'd be leading to heartbreak hotel. I think you have to remind yourself why many got into the game in the first place.
Once they've broken that taboo and got used to the lifestyle, it's very hard to give up. Most will never earn that coin again. Invariably when it goes Pete Tong, they go back on the game. I've seen it on Adultwork many, many times. And I suspect many feel the same way about punters too. That is, wouldn't even consider it. There will always be exceptions, of course.
Why would somebody even aspire to having a prostitute as a potential girlfriend, unless the idea of your woman getting fucked by all an sundry turns you on or if you have an open relationship? Otherwise, I don't get it. Big trust issues there, either way, if you're one who doesn't play away from home.
My advice would be to just have a great time, enjoy yourself and when done, get somebody who isn't a hooker because they're generally damaged goods. Sorry if that upsets people. Hard, but fair in my experience and based on many post punt chats in the flesh and forums. I find the mindset of a prossie on that aspect very interesting, some of the time. There are a small minority who I genuinely like, are intelligent to chat with and have something interesting to say. And who I also respect as people, but most I find are touts and manipulative bores.