To sympathisers, though, all of his crimes came under the general heading of misdemeanors.
If something comes under a particular authority, it is managed or controlled by that authority. They were neglected before because they did not come under the Ministry of Defence
If you come under attack or pressure, for example, people attack you or put pressure on you. His relationship with the KGB came under scrutiny
If something comes under a particular heading, it is in the category mentioned. There was more news about Britain, but it came under the heading of human interest