In most westernised countries there is a treaty called the Hague Convention which provides a set of standard rules that should be followed by every country that has signed up to the convention.
As a general rule, for married couples, if either parent wishes to take the child abroad for a holiday or premanently then they have to seek and obtain permission from the other parent. The issue is more complex where the parents are not married.
In family law, "Child Abduction" is the area of law covering children being removed from their normal country of residence:
Wrongful Removal - when one parent takes a child abroad without the permission of the other parent or the court.
Wrongful Retention - where a parent did obtain permission to take the child abroad temporarily, perhaps for a holiday or family visit, but then later fails to return the child on the agreed date.
The courts when dealing with child abduction matters are primarily dealing with which country's court should deal with the welfare arrangements for the child concerned. If, for example, a child has been brought to England & Wales without the permisssion of the other parent or the foreign court the normal result is for the child to be sent back to what is called their country of habitual residence.
There are exceptions:
if you can show that the other person agreed to the removal
if the child was spending equal amounts of time in two different countries
if there was significant violence which could not be controlled
if the child was of an age where they objected to a return and they are of an age where they are deemed mature enough to object.
The examples listed above are not exhaustive.
It is also important to consider how the foreign courts will deal with a returning child and parent. For example, a parent and child returning from the UK to the USA after being ordered to return after a finding of child abduction is likely to be treated in a significantly different manner than if that parent and child were returning from the USA to the UK as on the whole US courts take a more punitive view to a parent seen as an abductor than the courts in England & Wales do.
In an ideal world parents need to make the proper application to the court for a relocation of the child to the country they seek to live in rather than take actions which can lead to severe civil and criminal sanctions.