The Shulhan Aruch (Orah Haim 309:1) rules that it is forbidden on Shabbat to carry a young child who is holding in his hand an item that is considered Mukse (forbidden to move on Shabbat). For example, a ball is considered Mukse for adults, but is permissible to be handled by children. While a child holds a ball in his hand, it is forbidden for an adult to carry the child on Shabbat. The same applies if the child holds in his hand a pen, a coin or a dollar bill. If a child holds a Mukse item in his hand, then an adult who carries the child is considered as though he carries the Mukse item, and this is therefore forbidden.
The Shulhan Aruch adds that if the Mukse item is an object of value, such as if the child holds a $20 bill in his hand, then one may not even walk the child by holding his hand. The Sages were concerned that if one holds the child’s hand, and the valuable object drops, the adult may pick the item off the ground, in violation of Shabbat. They therefore forbade even holding a child’s hand while he holds a valuable Mukse object.
The Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939) writes that there is room to be lenient in this regard under extenuating circumstances. If, for example, the child is crying, frightened or for some other reason desperately needs to be held, and he refuses to put down the Mukse object, then one may carry the child. Under normal situations, however, one may not carry a child while he holds a Mukse object.
Summary: It is forbidden on Shabbat to carry a child while he holds in his hand a Mukse object, and if the Mukse object is valuable, one may not even hold the child’s hand. This is permissible only under extenuating circumstances, such as when a child is frightened and refuses to let go of the object.
