The earliest mention of the Amurru dates back to the Akkad Empire. The successor of Naram-Sin, Sarkalisarri, had a year of reign called "the year of victory over the Amurrum in the mountain of Basar." According to the researchers who looked at the location of this country in these Mesopotamian texts of the 3rd millennium BCE, the term generally refers to the maritime regions of the West. It is also the meaning that it seems to have in the archives of Ebla.
Here is a letter EA161 from Amarna which informs us that, in the next millennium, the city of Tunip was a place of residence of the king of Amurru: "My lord said, "You hid from Hani ". Let the gods and the Sun be witnesses: I swear I resided at Tunip. The king spoke of the reconstruction of Sumur. The kings of Nuhasse were at war with me and took my cities according to the advice of Hatip. So I did not rebuild it. Now, urgently, I will rebuild it. [...]" I think it must be understood that in Tunip the presence of Aziru, king of the Amurru, is due to his war against Nuhasse, which led him to neglect Sumur. EA 165 states that the Nuhasse is only two days' march from Tunip.
Today, the majority of researchers hypothesizes a location of Tunip on the Orontes. But, because of its probable earlier designation under the name Unipsum, I position it more towards the north. The city seems to me near Aleppo, and to the west of the Euphrates. This territory coincides with Yamhad and/or Mukish. Yamhad is a name of country before the 15th century BCE, however, I find that in the time of Aziru, Mukis and Amurru appear to be one and a same whole. Probably the Mukish was an Assyrian name of the country that was called Amurru by the Mesopotamians. In the texts of Ougarit, the two designations are present: Mukish is rather used to designate the terrestrial neighbor, whereas the Amurru seems to be a set of territories larger around the Mediterranean, especially known by its king. I think that a good part of the south coast of Anatolia was part of it, and that, looking more closely at the text of the statue of Idrimi of Alalakh and the letters of Amarna: Hawaliya or Wahliya appears to me as part of the Amurru from a long time, while Sumur, Ampi, Ullasa and Sigata are newly conquered port cities (click on these city names to see my localization proposals).
This is why, in the first millennium BC, the Assyrians continued to use the denomination "Mukis", deforming it into "Muski" (or Mushki), to designate the southwest Anatolia, from the Levant to Caria and Phrygia. Indeed, the Assyrian writings of the first millennium BC mention this opponent country, situated in the south-west of Anatolia, with King Mita at its head. Many historians have made the connection with the king Midas of Phrygia:
The hypothesis of a country of Muskhi equivalent to that of Mukish at a millennium of interval of time gives meaning to the annals of the Assyrian king Tukulti Ninip II, which from Huzirina, in two days of march, arrived until the mountainous regions of the Mushki.
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My notes of veracity :
From the middle of the second millennium BC, "Mukis", "Amurru" and Mira designated a same maritime country: 2,5/5
The Mukish and the Mushki are two names of the same country, the first in the 2nd millennium BC, the second in the 1st millennium: 2,5/5