"The closest comparison I can see would be to people born in the Louisiana Purchase Territory prior to 1803. They were born citizens of France, or wherever. Did they automatically become US citizens when the US purchased the property? Or did they remain citizens of their native country?"

This is what I was trying to describe. I would say that technically when the lands of the Louisiana Purchase became United States Territories those persons became provisional citizens of the United States. Technically I was taught in American Government class that a person could have dual citizenship if he was born in a foriegn country to American parents. In such cases he could be a dual citizens of both the country that he was born in and the United States at the same time BUT upon his 18 birthday he would have to declare which country he would in the future be a citizen of.

Even at that, in returning to the Louisiana Purchase. I have been told that under French Law any person born in a territory that had at any time been a territory of France, were still entitled to French citizenship. In other words any person, even today, who has been born in the states of Louisiana, Arkansas, Missouri, Oklahoma Etc. etc. etc. can claim under their law french citizenship with all due rights of any other French citizen.