There’s going to be more to this story than you are aware of. The procedures for getting a green card are the same as always, and though there may be higher scrutiny in background checks this wouldn’t affect the average person. Interviews don’t get cancelled for no reason - unless you know what category this person was applying under and their particular circumstances you can’t know what that reason was. (The fact that they are leaving the country instead of rescheduling the appointment leads me to think there was some issue with the underlying petition but like you I cannot actually know)
Tl;dr if OP’s family is on a normal family or employment based route to green cards there is no reason to worry other than the usual backlog/time delay issues.
Generally when you get a green card it is simply a case of submitting proof to the university and your status changes to domestic. It shouldn’t be any kind of red tape the way that proving a change in state residency etc is.