Or should they vote in their "home" elections?
This situation is akin to military members, who might live anywhere in the country. Military members have a "home of record", usually where they joined from, and that is the jurisdiction in which they vote by absentee ballot.
How is the college student situation sufficiently different from the transitory military member? Both live only temporarily in their current location, even if "temporarily" means 2-4 years. Is there a legitimate argument to be made in favor of restricting voting of college populations in the college towns, and instead requiring them to vote by absentee in the elections of the municipality they call "home"? For purposes of this discussion, let's set aside whatever laws and rulings are present and discuss the issue on its own merits.