In addition to the general considerations of (1) who is allowed to use non-deadly physical force to maintain order in a shop (which I think that one could do if "deputized" by the property owner or to protect the property of another as well), and (2) the use of non-deadly physical force to make a citizen's arrest (which many of these scenarios would justify as the disorder would be a crime if committed by an adult), (3) I suspect that there is also some point at which a bystander may intervene to prevent harms associated with an unsupervised minor being at large and in need of supervision.

Generally speaking, intervention with the minimum reasonable non-deadly physical force to prevent property damage, or an assault, or a threat, is going to be permissible. As to the third reason, for a mentally normal ten year old, that might be a stretch, for a four year old or a clearly impaired older child, it might not.

One could approach the child, say, "where's your mom or dad", "do you have a babysitter or sibling around?", "what is your name?", or "are you lost?" and detain the child under a satisfactory answer is provided or a suitable authority arrives, to prevent the problem of a child being lost, abducted or hurt by the child's own actions.

It would be quite hard for a parent, guardian or babysitter to complain about this kind of conduct when the child was unsupervised and is released as soon as you confirm that this really is a responsible adult or older minor who is responsible for the child.

It would be important in doing so to not secret away or isolate the child, to try to determine the location of the child's caretaker, to refrain from doing anything that would harm the child, and to seek assistance from an authority within a reasonable time. Typically, if no caretaker appeared, a cop would come and the cop would oversee the situation until a social worker could come.

For example, Colorado has the following statute that would apply once a cop arrived (omitting lengthy provisions that apply to newborn children):

> § 19-3-401. Taking children into custody
> 
> (1)	A child may be taken into temporary custody by a law enforcement
> officer without order of the court:
> 
> (a)	When the child is abandoned, lost, or seriously endangered in such
> child's surroundings or seriously endangers others and immediate
> removal appears to be necessary for such child's protection or the
> protection of others;
> 
> (b)	When there are reasonable grounds to believe that such child has
> run away or escaped from such child's parents, guardian, or legal
> custodian and the child's parents, guardian, or legal custodian has
> not made a report to a law enforcement agency that the child has run
> away from home; . . . 
> 
> (1.3)	A child shall be taken into temporary custody by a law
> enforcement officer without order of the court when there are
> reasonable grounds to believe the child has run away from the child's
> parents, guardian, or legal custodian and the child's parents,
> guardian, or legal custodian has made a report to a law enforcement
> agency that the child has run away from home.
> 
> (1.5)	An emergency exists and a child is seriously endangered as
> described in paragraph (a) of subsection (1) of this section whenever
> the safety or well-being of a child is immediately at issue and there
> is no other reasonable way to protect the child without removing the
> child from the child's home. If such an emergency exists, a child
> shall be removed from such child's home and placed in protective
> custody regardless of whether reasonable efforts to preserve the
> family have been made.
> 
> (2)	The taking of a child into temporary custody under this section
> shall not be deemed an arrest, nor shall it constitute a police
> record.

A child is considered neglected or dependent under circumstances including the following (provisions related to drug or alcohol abuse by parents omitted):

> § 19-3-102. Neglected or dependent child
> 
> (1)	A child is neglected or dependent if:
> 
> (a)	A parent, guardian, or legal custodian has abandoned the child or
> has subjected him or her to mistreatment or abuse or a parent,
> guardian, or legal custodian has suffered or allowed another to
> mistreat or abuse the child without taking lawful means to stop such
> mistreatment or abuse and prevent it from recurring;
> 
> (b)	The child lacks proper parental care through the actions or
> omissions of the parent, guardian, or legal custodian;
> 
> (c)	The child's environment is injurious to his or her welfare;
> 
> (d)	A parent, guardian, or legal custodian fails or refuses to provide
> the child with proper or necessary subsistence, education, medical
> care, or any other care necessary for his or her health, guidance, or
> well-being;
> 
> (e)	The child is homeless, without proper care, or not domiciled with
> his or her parent, guardian, or legal custodian through no fault of
> such parent, guardian, or legal custodian;
> 
> (f)	The child has run away from home or is otherwise beyond the
> control of his or her parent, guardian, or legal custodian;
> 
> (g)	The child tests positive at birth for either a schedule I
> controlled substance, as defined in section 18-18-203, C.R.S., or a
> schedule II controlled substance, as defined in section 18-18-204,
> C.R.S., unless the child tests positive for a schedule II controlled
> substance as a result of the mother's lawful intake of such substance
> as prescribed.

I don't have easily at hand legal authority authorizing a third-party who is not a law enforcement officer to take custody of a dependent or neglected child until a law enforcement officer arrives, but I strongly suspect from the context that this is allowed either under common law, or a statute that I have not located, or some legal fiction (e.g. that the citizen is impliedly deputized by the law enforcement officer after the fact).