I was planning to leave this for someone else to answer, but it looks like the question may be closed by people who misunderstand it.
Our resident expert on UK immigration matters, Gayot Fow, has responded to a comment in which I said:
long-term visas will generally only be granted after the traveler shows "performance" (i.e., compliance) with a number of "normal" (i.e., not long-term) visas. I do not know what that number is.
@phoog this is right. the actual criteria that you do not know is classified and there's no reason to know it. I don't know it either. But thinking along the lines of 5 - 7 visitor visas jives with personal experience. Nobody gets to see what's classified. You should promote your comment.
He later added:
In addition to phoog's comment, you do nothing special other than tick the long-term box. If they think you qualify they will issue it. Otherwise they will not issue it, there are no special documents to add and there is nothing extra to do except tick the box.
The advice, then, seems to be that you request a long-term visa by ticking the box. To maintain your credibility, your application should presumably include a brief description of the circumstances that prompt you to ask for a long-term visa.
You should also consider it likely that you will be given a normal short-term visitor visa instead of a long term one, unless you have already held a number of short-term visitor visas. If this happens, you should continue to ask for long-term visas in your subsequent applications, and it is likely that you will eventually be successful.