(for this match, we're using the film versions of the Harry Potter universe, and the film/game/IDW versions of the 1984 Ghostbusters)
Due to being the Muggles' top experts in the supernatural, the Ghostbusters get an invitation by Dumbledore as part of his new cultural exchange program. This friendly trip coincides with Harry Potter's first year. The Ghostbusters have an open invitation throughout all the films.
Egon gets a fellowship to conduct research on the history and "mechanics" of magic. Ray will work on an extensive collaborative engineering project merging magic and technology. Winston will get to study mysticism of ancient Egyptian culture. Peter is tagging along because he told Dumbledore that he was their "manager." They have no prior knowledge of this world (otherwise, what's the point of an exchange?) but both Dumbledore and the Ghostbusters have good intentions for this visit. The Ghostbusters have brought over all their equipment and are eager to get to work.
However, everything else in the movies remains the same -- the looming schemes of Voldemort, the threats of the Death Eaters, the destiny of Harry Potter himself. How do their respective worlds change? How do the boys fare? Which HP characters do they get along with the best? Or the least? Can they lend a hand whenever shit hits the fan at Hogwarts? And more open ended questions!