Six-year-olds that I've known seem to be at a stage when they tend to cry fairly easily. I've always assumed it's because they're still pretty young, and it's still easy for them to get overwhelmed and/or to feel "emotionally solid enough" to be able to deal graciously with some of the things that bother them.
Also, they're old enough to have developed a certain type of pride and to have a threshold for humiliation that a younger child may not have. Children always have whatever they have that goes on between them and their mothers, but then there's how they want other people who are bigger/older than they are to see/treat them.
If he particularly admires family members who are older than he is, or if particularly wants their approval and respect and/or if they're not very understanding or appropriate in their response to catching him do something wrong; these kinds of things could explain why he cries.
Children this age often have tremendous admiration for, say, their father or for older siblings.
Mothers often know how not to make a child feel too horrible about doing wrong, so young children can kind of get used to having that relationship with the mother that involves her seeing him "at his worst" and both of them getting past it.
Or some mothers make such a habit out of acting as if the child is always doing wrong, the child may stop caring so much about even trying to have her approve of/admire what he does). If a mother is always scolding a child (and I"m not suggest you are - only saying it's what some mothers do), the child kind of tunes it out and doesn't take it all that seriously.
Another scenario might be that a young child whose mother frequently catches him doing things wrong may be able to deal with at least that one adult catching him; but if it gets to the point where it seems as if "everyone" in the family is "on his back" and "judging him" or "having an opinion" about what he does; that could potentially feel like "too much" for a young child (and six is still very young).
I'm not an expert, but the fact that he crouches over and cries for so long suggests to me that he isn't particularly crying over the simple act of someone's catching him. It seems to me that he's most likely feeling as if "everyone's on his back" and maybe he's feeling as if it's all just too much to deal with.
His father is one thing, but I'd tell all other relatives to tell me if they caught him doing something, and I'd deal with it.