My wife is Chinese, and her sister’s kids (and their Chinese cousins on their father’s side) all chew with their mouths open. Most Chinese people (Chinese by culture, not by race) chew with their mouths open. In their culture, as I’m sure you know, it is not improper to do so.
I’ve tried to get my wife to point out to her sister that in our culture, chewing with your mouth open is improper and that using the proper fork and saying please and thank you will all be nullified and then some if they chew with their mouths open. (They don’t just chew with their mouths open; they suckle and gurgle and smack and make all kinds of noises, as do lots of Chinese. It’s not that they’re rude; it’s just that it’s not considered rude in their culture). And if they’re not taught that that’s not acceptable in America, they’ll never learn different, and they’ll never know why nobody invites them to lunch.
Anyway, I figure that since you’re so concerned about manners, you probably know to teach your children to chew with their mouths closed. But I thought I’d mention it just in case.