You may be right.
Usually, when a kid is a freshman/sophomore, there are issues with "academic progress", and the fact one school's "pre-requisite" courses are not the same at another school.
But when guys are really really close to graduation, they can usually take some "final" courses at the new school and then earn the diploma at the old school. This is very common in law school, you will often see a 3rd year student "transfer" to a school where he/she wants to live/practice, but the final coursework qualifies as the completion of the J.D. degree at the first school. If you "transfer" after your first year in law school, you get your degree from your new school, but if you transfer after your second year in law school, you generally get your degree from your original school.
Who knows, maybe things have changed over the past 30 years.