Can a child 4 and half be taken away to another county from their parent who only has partial custody?

Usually they can, as long as it is within a reasonable distance so as not to preclude visitation. But the moving parent will be responsible for at least half of the travel expenses,(if not all.) You will need to gage weather it is worth paying a lawyer to contest it. But you also have a right to know where your child is living, and with whom.

What do you mean by "partial custody"? How is the order/judgment worded? If you have joint legal and physical custody, then the moving parent has to get the court's permission. Courts do not like to have a parent effectively taken out of a child's life. If the other parent has sole custody and you have a substantial time share, then the other parent will have a much better chance. And WHY is the other parent moving to another country?

If your child doesn't have a passport, yet, then he can't leave the country without both parents consent. Newer passport laws require both parents to be present to process a child's passport application. There is a work around (notorized power of attorney), but they really want both parents to be there and sign the paperwork. I think it stemmed out of some spousal abductions in the 90s.

Also, that only applies to US children. Don't know about other countries.