When did the Assyrian identity disappear? Babylonian?

While you wait for more targeted answers, you might be interested in reading some previous threads on the Assyrians:

If there are still people who identify their ethnicity as Assyrian, are there also still Babylonians or other Ancient Mesopotamian ethnic groups today? If not, why did Assyrians survive but not others? Are they really Assyrian? - 69 comments, over 5 months old.
The topmost comment defines Assyrian identity in religious terms and uses that to talk about the journey of the Assyrians and other Near Eastern Christians from Christianity's foundation into modern times.
Were the Assyrians victim to a culture-wide damnatio memoriae? - 7 comments, over 2 years old.
This thread goes into the destruction of the physical aspects of Assyrian culture from the capital to the temples and art.
The AskHistorians Podcast - Episode 19 Discussion Thread - Assyrian State Archives - 23 comments, over 4 months old.
The comments provide translations of ancient Assyrian records and the podcast goes on to discuss the state of historiography on the Assyrians.