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In the Harry Potter universe, is there any mention of an exchange student at Hogwarts or one of the Hogwarts students going on exchange at another school?
This happens all the time in the "muggle" school systems I'm sure it would happen in the wizarding schools as well.
In a word, Yes. In Chapter 7 of Goblet of Fire (Bagman and Crouch) we learn that there are inter-school "exchange trips" in the wizarding world. Unfortunately the Weasley family, being as poor as mud, were unable to afford the expense:
‘Who d’you reckon they are?’ he said. ‘They don’t go to Hogwarts, do they?’ ‘’Spect they go to some foreign school,’ said Ron. ‘I know there are others, never met anyone who went to one though. Bill had a pen-friend at a school in Brazil … this was years and years ago … and he wanted to go on an exchange trip but Mum and Dad couldn’t afford it. His pen-friend got all offended when he said he wasn’t going and sent him a cursed hat. It made his ears shrivel up.’
Harry laughed, but didn’t voice the amazement he felt at hearing about other wizarding schools. He supposed, now he saw representatives of so many nationalities in the campsite, that he had been stupid never to realise that Hogwarts couldn’t be the only one. He glanced at Hermione, who looked utterly unsurprised by the information.
In Harry's 4th year - 1994 to be precise - the Ministry of Magic revives the Triwizard Tournament. Cribbing from that wiki article, it is a wizarding tradition dating back to 1294 and held every five years, with Hogwarts in England, Beauxbatons in the French Pyrenees and Durmstrang in Norway or Sweden rotating the hosting duties. Hogwarts students matriculate after 7 years of school and Beauxbatons students take their O.W.L.s after six years instead of five; assuming Durmstrang students follow a similar course of study, then every student to have graduated from any of those schools between 1294 and 17921, would have spent part of a year abroad, studying in the halls or on the grounds of a foreign institute.
Dumbledore gives significant speeches to the assembled students, as is his custom, one at the beginning of the year announcing the tournament and one after the conclusion. One of the themes he touches upon is the importance of building international ties, and particularly bonding and standing with your friends during trying times. As Exchange Student World states, the top 5 reasons for foreign exchange programs are:
Aside from learning a new language2, these goals line up almost exactly with the expressed and observed goals and occurrences of the Triwizard Tournament in Harry Potter and the Goblet of Fire. That makes it the grandest, most elaborate and most extensive foreign exchange program in the wizarding and muggle worlds.
Oh, and aside from the dying as well.
1: The tournament was canceled after the 1792 tournament due to injuries. This is apparently a canon quote from HP&tGoF, but it is admittedly an odd year, proving that the tournament was not consistently held every five years. If it were, there would have been tournaments in 1789 and 1794, and none in 1792.
2: In typical fashion, every Durmstrang and Beuaxbatons student featured in both the book and the movie have a superior grasp of the English language than any English speaking school student has over another language. This is partially a storytelling conceit, as it allows JKR to keep the text of the books in English, Parseltounge or magic. But it also agrees with my limited experience with foreign exchange students visiting my own English speaking schools: nobody learned much language from their hosting duties and the exchangees primarily learned English slang and pop culture.
Exchange programmes exist
Adding to Valorum answer in Pottermore Rowling describes the existence of an exchange programme with the brazilian wizarding school of Castelobruxo:
Castelobruxo students wear bright green robes and are especially advanced in both Herbology and Magizoology; the school offers very popular exchange programmes for European students* who wish to study the magical flora and fauna of South America.
Also, on the Train to Hogwarts, in tHBP, when Harry is eavesdropping on Malfoy, Malfoy mentions the fact that his father had wanted Draco to go to Durmstrang, but his mother didn't want him to be so far from home, so he went to Hogwarts instead. This means that it is possible for wizards to go to schools in different countries, at the very least. this is as close as i can find for something slightly relevant to the question that hasn't been said yet
"Father actually considered sending me to Durmstrang"
harrypotter.wikia.com/wiki/Durmstrang_Institute
