Is it possible to study abroad as a PhD student?
I don't think what you're describing is all that common, and I've never heard of any sort of systematic program, but it does happen.  I ended up spending two months abroad during grad school to participate in a once off trimester program in my area of specialty.  There weren't any standard arrangements in place, though: my home university (very generously) moved things around so I could do it.  I think I've heard of other cases, all similarly involving unique circumstances like faculty moving around or particular personal connections.