It is asked, whether the children born of citizen in a foreign coutry are citizens? The laws have decided this question in several countries, and their regulation must be followed. By the law of nature alone, children follow the condition of their fathers, and enter into all their right; the place of birth produces no change in this particular, and cannot of itself furnish any reason for taking from a child what nature has given him; I say "of itself," for civil or political reasons, ordain otherwise.
naturally, it is our extraction, not the place of our birth that gives us rights;
Settlement is a fixed residence in any place with an intention of always staying there. A man does not then establish his settlement in any place, unless he makes sufficiently known his intention of fixing there, either tacitly, or by an express declaration. However, this declaration is no reason why, if he afterwards changes his mind, he may not transfer his settlement elsewhere. In this sense, a person who stops at a place upon business, even though he stay a long time, has only a simple habitation there, but has no settlement. Thus the envoy of a foreign price has not his settlement at the court where he resides.
The natural or original settlement is that which we acquire by birth, in the place where our father has his; and we are considered as retaining it, till we have abandoned it, in order to chuse another. The acquired settlement (udseititium) is that where we settle by our own choice.
Vagrants are people who have no settlement. Consequently those born of vagrant parents have no country, since a man's country is the place where, at the time of his birth, his parents had their settlement, or it is the state of which his father was then a member; --which comes to the same point: for to settle for ever in a nation, is to become a member of it, at least as a perpetual inhabitant, if not with all the privileges of a citizen. We may, however, consider the country of a vagrant to be that of his child, while that vagrant is considered as not having absolutely renounced his natural or original settlement.
What I gather from Vattel's words is that the he would agree a father who is British produces British children who have all the citizenship rights of their father. The children are British. For the children to have any citizenship other than their father's, the children would have to abandon (to renounce) their natural citizenship to acquire another citizenship.
Had the father been a permanent settler in another land, then it is natural his children would have the citizenship of their father.
By nature, a child's original citizenship can be only with the nation of his father's citizenship, unless the father is a vagrant.