I am uncertain if I understand your question correctly.
The Modular Travel Card is personal and can solely be used by the person registered, and it is not transferable. Therefore, it is either registered in your name or with one of your children, and only this person can travel with it. The person travelling does not need to be accompanied.
Your children can travel for free with you providing you hold the Modular Card and the Junior Card; is the latter not the case, your child needs a half fare ticket or a Children Day Pass when travelling with you, same as when your children are travelling alone.
Please kindly rephrase your question if this answer is not what you were looking for.