No, their college won’t matter as much as their job experience and performance. (In fact, their “performance” in college probably won’t matter at all at that point.) However, it can have an indirect effect.
When you’re looking for your second or third job out of college, most employers are going to care far more about your experience at the first and/or second job than where you went to college. Where you went to college becomes less relevant, because your performance at the former job is much more indicative of how you will perform at the next job than where you went. However, if your college influenced your first job - like let’s say the job asks for or prefers experience at a top strategy consulting firm and your Cornell BS helped you get that kind of job - then it obviously has an indirect effect.