In the earliest Sumerian sources concerning the Amorites, beginning about 2400 BC, the land of the Amorites ("the Mar.tu land") is associated not with Mesopotamia but with the lands to the west of the Euphrates, including Canaan and what was to become Syria by the 3rd century BC, then known as The land of the Amurru, and later as Aram and Eber-Nari.  They appear as an uncivilized and nomadic people in early Mesopotamian writings from Sumer, Akkad, and Assyria, especially connected with the mountainous region now called Jebel Bishri in northern Syria called the "mountain of the Amorites". The ethnic terms Mar.tu (Westerners), Amurru (likely derived from 'aburru', pasture) and Amar were used for them in Sumerian, Akkadian, and Ancient Egyptian respectively. From the 21st century BC, possibly triggered by a long major drought starting about 2200 BC, a large-scale migration of Amorite tribes infiltrated southern Mesopotamia. They were one of the instruments of the downfall of the Third Dynasty of Ur, and Amorite dynasties both usurped native rulers of long-extant Babylonian city-states such as Isin, Larsa, Eshnunna, and Kish and also established new ones, the most famous of which was to become Babylon, although it was initially a minor and insignificant state.  Known Amorites wrote in a dialect of Akkadian found on tablets at Mari dating from 1800-1750 BC. Since the language shows northwest Semitic forms, words and constructions, the Amorite language is a Northwest Semitic language, and possibly one of the Canaanite languages. The main sources for the extremely limited knowledge about Amorite are the proper names, not Akkadian in style, that are preserved in such texts. The Akkadian language of the native Semitic states, cities and polities of Mesopotamia (Akkad, Assyria, Babylonia, Isin, Kish, Larsa, Ur, Nippur, Uruk, Eridu, Adab, Akshak, Eshnunna, Nuzi, Ekallatum, etc.), was from the east Semitic, as was the Eblaite of the northern Levant.

Answer this question "Where did the Amorites originate" by extracting the answer from the text above.
In the earliest Sumerian sources concerning the Amorites, beginning about 2400 BC, the land of the Amorites