I never carry my PP around with me in Shenzhen. I've encountered a few ID checks over the years in the metro and when they see my foreigner face, I get waved right through. I do have my NY drivers license in my wallet if they really wanted to see something. It seems they are more concerned about ID checking citizens than foreigners in the past, I don't know if that is changing. If I were an ethnic Chinese foreigner (i.e. looked Chinese but not) I may be more inclined to carry around some ID or photocopy proving my citizenship.
Did you get your children a passport card in addition to regular passport? If so it may be useful to carry that around instead of the regular passport. I wouldn't want to risk losing a regular passport with my visa inside as replacing it would be a hassle.