Most good universities allow students the flexibility to study elsewhere. This is an extremely useful to the students who take advantage of this. No single university can claim to have the best professors in all disciplines. So the student can go and study a few courses from another place where better teachers or facilities are available. This does not reduce the value of the parent university, as they are now giving a degree to the student who is perhaps better prepared. This will further enhance their own prestige and value.
Such a flexibility is also useful to students who may have personal reasons to study in another location. For example, if one is having some medical problems where support from family is needed, then the student can spend a semester in a college in the hometown, knowing that the credits for these courses will be given by the parent university.
By having such a flexibility, universities can attract students from other places to spend a semester, and experience the life at their campus. It may be useful to attract them later for post-graduate programs.
Therefore, universities should have mechanism to admit temporary students for 1-2 semesters, and issue them transcripts for the courses done. And also, allowing their own students to go out for 1-2 semesters, and do credit transfer/waiver for courses done elsewhere.