Thank you, Uncle Jack. I'm wondering what "do" actually means there? Doesn't it have a sense?It doesn't refer to any specific action, but to how they are getting on with life generally. "Do" is the usual verb with "fine", although it is often omitted, as it is in B's initial response. In any case, "be" and "do" are often interchangeable in conventional greetings, and a "be" question ("How are you?", for example) might be answered with a "do" answer ("I'm doing fine") or a "do" question ("How are you doing?") might be answered with a "be" answer ("I'm well, thank you").
A's sentence is wrong. It should be "How are your children?" because "children" is plural.