The oath of the people of Ammon was a very specific one, taken by a particular set of people. The oath did not, and was not intended, to cross generations (for an analysis of the nature of this oath, see the commentary following Alma 24:11). These sons were likely less than age 8 when the oath was taken in the land of Nephi, and now at least ten years have passed. Thus some of these sons might be older, but many would be in their early teens. These young men take up the arms that their parents could not.
The last phrase is interesting in that it indicates that these young men “called themselves Nephites.” Their parents are called the people of Ammon, and perhaps maintained some connection to their identity as converted Lamanites. Their sons, however, have spent most of their life in their new home, and their self-definition is not Lamanite, but Nephite. This is yet again an indication that the terms Lamanite and Nephite pertain more to the political sphere than to the genealogical one.