It depends on the exact language in the custody decree, but ordinarily a custodial parent is allowed to place a minor child in the temporary care of any responsible adult.
The non-custodial parent can file for custody.
Leaves them permanently or in violation of court ordered custody/visitation time? In either circumstances, the non-custodial parent may file an action for contempt of court against the custodial parent and/or file for custody/visitation modification based on the same.
Maybe.
Only with a modification. see link
no need to, sooner or later they all f..k up and get them selfs in trouble again
An emigrant is a person who leaves their country to settle in another country.
That depends on state law where the child legally resides (or country if outside the USA). Every place has a different statute or guideline regarding what constitutes abandonment and in all cases, any extenuating circumstances are taken into consideration before making such a ruling.
With the permission of the residential parent? Yes. see related question.
The non-custodial parent has the right to file a motion against the custodial parent based on the same. As a result, the non-compliant parent may be held in contempt of court. If they are, penalties may include a monetary fine, jail or modification to the original custody order.
An expatriate or expat is a person who leaves their own country to live and settle in another country.
immigrant
The one who leaves, as the one who stays not necessarily knows you are going somewhere.