A short guide to iść/chodzić:
iść - something that happens right now. "to be walking" and "to be going" (on foot)
chodzić - something that happens regularly, habitually. "to walk" and "to go" (on foot)
Also, if there is no direction specified, if it is just 'walking around', "chodzić" also works for "to be walking".
They could say that. They could also say "My kids are walking." This is in the sense that they are able to walk. In fact, it's more common to use "walking" in this particular case. Also idioms like "walking to and fro", that is in a couple of directions without going anywhere.
I see it more as a sentence fragment, where we haven't learned all the words yet to make a better sentence. So, yes, I see this as getting ready for "moje dzieci chodzą ____". Though I can think of a few cases where a parent would say "my children walk" : they don't take the bus to school, they walk; they don't race around the library like some children, they walk.
Firstly, both refer to walking (going) on foot.
"iść" is for Present Continuous. "to be going" (on foot), "to be walking".
"chodzić" is for Present Simple. "to go" (on foot), "to walk".
However, if there is no destination specified, it it's just walking around, "to be walking" is also "chodzić". So it would work fine in this sentence, because there is no destination.