I'm interpreting 'old students' in this context more as 'students who are established fixtures of the gym' as opposed to 'students over a given age'. Talking with them. Be honest about the reasons you want to ask them and figure out if it's something that can change or if it really is just best for them to leave. If it's because they're not getting from the gym what they want and you can't offer it, being upfront that it's probably time for them to find a gym that does offer what they're looking for isn't a pleasant conversation, but can be a necessary one. If it's because they've done something inappropriate, tell them to get out the same way you would anyone else who did the thing. If it is the former, be clear that they're still welcome to come back for stuff like open mat, seminars, etc, but that you're just not looking to change the gym to provide the specific thing - try to keep it on good terms. One thing to consider is 'damage control'. Whenever you ask someone to leave there's a chance for rumors to start up. When it's a long established student it's super likely others hang out with them outside the gym and/or talk with them; rumors are almost a guarantee, especially if they leave on bad terms. Being upfront about why [insert name here] isn't at the gym anymore can help how people feel about it so they're not worried they're about to be next for what feels like no reason.

Thanks. I meant how do I remove them from the list of students on beltchecker. I can reject them, but not remove them from the list of students. Your advice is great though!