Migration of Semitic peoples took place already in the Early Dynastic Period. The Semitic peoples merged with native population and adopted Sumerian culture, religion and form of government. Lugal-zage-si failed to establish strong and centralized Sumerian state under his authority what was taken advantage by Sargon of Akkad who came to power by overthrowing the king of Kish in BC. Sargon established his capital in the city of Akkad which has not been located yet , defeated Lugal-zage-si, conquered all Sumerian city-states and established imperial authority over Sumer.
History of Mesopotamia
Babylone (civilisation) — Wikipédia
The Sumerians had a rather advanced civilization with their own elaborate language and system of writing. They also had extensive knowledge about our solar system, astronomy and mathematics. We still use the same mathematical, time and calendar systems created by the Sumerians today. In addition to their advanced language and understanding of the cosmos, the Sumer civilization also used advanced farming techniques. Much like their elaborate language system, the Sumerians also had a complex religious system with hundreds of gods, cosmology and rites. According to their texts, each city in the civilization was guarded by its own god.
From the end of the seventh century BC when the Neo-Assyrian state fell to the mid-seventh century AD, it survived as a geopolitical entity,    for the most part ruled by foreign powers such as the Parthian  and early Sasanian Empires  between the mid-second century BC and late third century AD during which a number of independent Assyrian states such as Adiabene , Osroene , Beth Nuhadra and Beth Garmai arose. The final part of this period saw Mesopotamia become a major centre of Syriac Christianity and the birthplace of the Church of the East and Syriac Orthodox Church. A Semitic -speaking realm, Assyria was centred on the Tigris in Upper Mesopotamia , in modern terms, northern Iraq, northeast Syria, and southeast Turkey.
Before we go any further, a cautionary note regarding timelines of the period. There is general agreement amongst scholars of timing of names and events but absolute dates and how they are applied does vary. I have used a commonly accepted timeline and all dates are given that way. Therefore you will most likely see variation from some sources but even then the event is generally placed in the right context, so it is still identifiable and understood.