The sentence is examining a grammatical or contextual issue related to the words highlighted. The word "their" suggests a plural subject, while "New York City" is a singular noun. The correct possessive pronoun should be "its" instead of "their." Thus, the corrected sentence would read:

"For most of its history, especially since the 1860s, New York City has been undergoing major ethnic population changes."

So the correct option would be to change "their" to "its." If you're looking for the correct answer among the provided options, it likely corresponds to a missing option that addresses this issue.