If my child “sits down” in the middle of a Shabbos walk and refuses to continue, am I permitted to carry him home?

The answer to this frequent scenario will greatly depend on where it takes place. Obviously we are not talking about a case where there is a kosher eiruv, because in such a case he may be picked up and taken home. Likewise in a life-threatening situation, he may be lifted and taken home.

If he is in a reshus harabim (a public domain where carrying four cubits is biblically prohibited) you are forbidden to carry him. You will have an opportunity to practice your coaxing/ threatening/ pleading abilities, because there is not much else you can do. If he leans on you while walking in a way that one foot is always on the ground, it is permitted as long as you do not drag him.

If he is in a carmelis (a public domain where carrying four cubits is rabbinically prohibited) carrying is likewise prohibited, but you may ask a non-Jew to carry him home. [1]

If my child is carrying a stone in his hand, may I lift him up? Is it as if I am carrying the stone as well?

The Shulchan Aruch [2] teaches us that if a child is yearning for his father, the father may pick him up even though the child is holding a stone in his hand. However, it states, that this is on condition that the child would become ill if not lifted up. The reason it is otherwise forbidden is because it is considered as if the father himself is holding the stone. If the child would not become ill, even though the child is yearning for his father it is forbidden to lift him as long as he is holding muktze.

Tosefos in Shabbos 142a asks that one should first make the child discard the stone and only then pick him up. Tosefos answers that the child would cry if he were made to discard the stone.

In other words we have a chain reaction: the child may be lifted while holding muktze if he would become ill, and the muktze need not be discarded if the child would cry.

Why is it considered as if the father carrying the stone, the child is?

Although it seems to be a classic case of tiltul min hatzad (carrying indirectly), however, the father is content for his child's sake that he is carrying the stone.

What if he is carrying money?

The Shulchan Aruch continues that if the child is holding money [3] it is forbidden to pick him up even though the child would become sick, because if the money would fall from the child’s hand the father would stoop and pick it up, which involves handling muktze directly. In contrast, if the stone would fall from the child’s hand the father would not be concerned enough about it to pick it up. (It does not mean that one must stand idly by while his child becomes sick, it means that lifting him with money in his hand is forbidden and it is up to the parent to either make him discard the money or substitute it for something else).

Rashi holds that even holding the child’s hand is forbidden when he is holding money lest the money falls and the father will pick it up, but the Ramban disagrees and permits it.

The Elya Raba (quoted by the Bi’ur Halacha) says that the Ramban can be relied upon where a case of illness is involved.

If holding his hand is forbidden, maybe I should not walk next to him either? What is the limit?

Let us answer that with another example.

Halacha says that one may not smell edible fruits on trees on Shabbos lest one picks and eats it. Why then are you permitted to walk within four amos of a fruit tree, are we not concerned that one might pick fruit to eat?

The answer is, when one is involved with something, in a "moment-of-forgetfulness" one might perform a prohibited action, but when not involved with the item there is no reason to be concerned that one will forget it is Shabbos.

Consequently, when smelling edible fruits, since one is already bending over to smell the fruit, we are concerned that one might forget and remove the fruit. Likewise, when holding a child's hand and the child is holding money, Chazal were concerned that since one is "involved" with he child, one might "forget" and lift the money if it falls.