How to arrange an invitation letter for research visit in another university (for PhD student)?
In mathematics, in the U.S., since people in Paris are vaguely acquainted with me, some years ago I was asked to be a supervisor for an Ecole Poly undergrad's "senior project", more-or-less. I had spare time-and-energy at the time, and the people asking me were known to me, and certainly gave the impression that they were not just spamming. (The student was very good, and later won a prize for his paper, "back home".)

So I think if you can get your local faculty to contact people who know them, and can recommend you to them, it might work well. (But, sure, sometimes people are already over-committed, as I am currently, so I sadly had to decline a similar offer, although I knew it would have been quite fun... Just not enough time.)