First, who is “you?”
No one can punish (take action against) a person for disposing of their property without violating the non-agression principle, in which case the individuals doing and supporting the punishing are acting in the capacity of a State. In a stateless society, people would be “punished” by actions of others on the free market. The freedom to associate includes the freedom not to associate, and association includes exchange. It is impossible, from the view point of economics, to predict what the market outcome would be since it depends on the choices and valuations of many individuals, but if individuals want children to be protected they could make choices that would have the effect of maximizing child welfare. For instance:
A father kills his child. If this occurs on property he owns, then no direct action can be taken against him. That is, no individuals can go onto the father’s property and arrest him, demand that he perform various actions such as consenting to have himself tried in a court, and so on, unless the father consents to allow such actions be taken against him. To move preemtively against the father would violate the NAP, it would constitute a usurpation of authority away from the individual (the father).
However, as fas as the father is a member of society, he is involved in market interactions with others. Thus he relies on others to sell him products and to buy his products. As knowledge of his action diffuses throughout the community, his avenues to transact may be cut off and he may be thus penalized directly. He may eventually submit to the demands of others that he perform actions in order to right the wrongs that they accuse him of, so that he may eventually be allowed back into society. Or, unable to right his wrongs in the eyes of his fellow men and unable to endure their scorn and unable to subsist because of their unwillingness to associate, he may impose exile upon himself.
Or, since in a free society every individual would likely need to cross the privately owned properties of many individuals in the course of everyday life, the father would find that he is not able to leave his property. The owners of the commonly used thoroughfares (roads, sidewalks and the like) are paid to produce a product that conforms to the demands of the consumers, or else they quickly lose business to competitors. As soon as consumers of the thoroughfares hear that Mr. X has killed his child, they may pay to keep Mr. X off of any roads they use. But it really works in reverse. One road bans Mr. X and so consumers prefer that road or thoroughfare, and the owner of that road profits by anticipating the wants of the consumers. Other roads follow, seeing the profit being made by the other road owners by banning Mr. X. Eventually, Mr. X is hampered from going anywhere, to the extent that consumers wish to keep him off of the roads that they are paying to use.
At the same time, consumers who are concerned with the plight of children who may be neglected by their parents could voluntarily pool their resources and form a corporation that could buy children from neglectful parents. Or, they could offer on the market, contracts between the corporation and parents where the parents agree to satisfy certain requirements of the child in exchange for money.
The explanation for how Mr. X would be “forced” to go to court/prison and make restitution of some kind follows just the same logic as above. Certain easements may be allowed. Thus, the owners of various roads, while banning Mr. X from using their roads for conducting business, grant easements for his travelling to a prison/courthouse where he may voluntarily submit to a trial. If this is what consumers want, they patronize the businesses that are participating in creating this product. The product being created is “allowing Mr. X passage to prison and nowhere else.” Of course, if Mr. X no longer wants to be in prison he can leave at any time, but the community can make his life as difficult as they choose, so long as they do not initiate force against him. The market could similarly pressure Mr. X to sign a contract agreeing to take actions to right his wrongs, or to perform certain labor, etc.