The frame
The Met's timeline page for Mesopotamia, 1000 BCE-1 CE, dates the "Neo-Assyrian empire" about 911-612 BCE. The page's events follow its capitals: in 879 BCE Ashurnasirpal II, who according to the page reigned 883-859 BCE, moved the capital from Ashur to Nimrud, ancient Kalhu; Sargon II, about 722/721-705 BCE, built a new capital at Khorsabad, ancient Dur Sharrukin, which took seven years to build; and his son Sennacherib, about 704-681 BCE, rebuilt Nineveh as his capital. Source 1.
Boundaries and overlaps
In the same list, with no north or south headings, a separate Babylonian sequence appears beside the Neo-Assyrian empire: the "Neo-Babylonian period", about 1000-539 BCE, and within it "Assyrian rule", about 750-625 BCE, followed by the "Neo-Babylonian kingdom", about 625-612 BCE, and the "Neo-Babylonian empire", 612-539 BCE. Assyrian rule thus appears as a chapter within the Babylonian sequence, and the Neo-Babylonian kingdom begins before the end of the Neo-Assyrian empire. The site does not copy the Assyrian frame onto the south: the Babylonian sequence belongs to its own Neo-Babylonian period. Source 1.
On this site
Three articles and one object are placed in this period. "Kalhu (Nimrud): the palace of Ashurnasirpal II" looks at the move of the capital in 879 BCE and at how the palace reliefs and inscription present the king. "Dur Sharrukin (Khorsabad): Sargon II's new capital" looks at a capital built over seven years, and at what the source does not say about why it was built. "Ashur: a first capital across six centuries" explains why Ashur's time as capital, which UNESCO dates from the 14th to the 9th centuries BC, is placed only here: it is the only phase the sources date within a single period. The object is the "Ivory plaque from Nimrud with two sphinxes", which the museum dates ca. 9th-8th century BCE and assigns to the "Neo-Assyrian" period. Sources 1, 2, 3.