The land of the two Rivers is divided into upper and lower Mesopotamia, the Zagros mountains and the slim strip of desert west of the Euphrates, oh and Elam. Sadly this wasn't the focus of the devs so units are a bit scarce. I know mounted troops were not really a thing outside of scouts but I think they are fun and good for gameplay, so I left them in, BUT limited “Cimmerian”(lol) horsemen to non-river food regions So they should pop up less commonly. Ahlamu and Arameans are also present but covered in the Canaan section. Akkadian influence can be felt up to carçemish.
-Akkadian barrack
-Akkadian range
-Akkadian conscription
-Assyrian stable
-Kassite stable
-Nairi(urartu) barrack
-Horse nomads
-Elamite range
-Elamite barrack

Mesopotamian factions can build a central archive in one of the big cities of Mesopotamia to reduce administrative level. This building has four upgrades gradually increasing the effect. It also reduces the global food and gold upkeep of units. It only functions in High Influence territories and will grant a lot of XP to Enemy generals as well as a hefty Legitimacy malus if damaged.

Hanigalbat
The Assyrians thrive on conquest, every (outpost) monument they erect abroad reduces their military upkeep. This is a bit tricky because the UI does not tell you, but on high Influence it just boosts Legitimacy. IF the Influence is low however it instead reduces global upkeep by 10% which may not be much, but the costs are microscopic compared to the other buildings so this bonus hits much earlier.
-Assyrian range
-Assyrian barrack
-Assyrian officer estates, Cavalry in minor food/plains settlements
-Mitanni horse trainer, Elite Cavalry in major settlements

Babylon
To keep up with other empires Babylon's slaves don't count towards the administrative burden. These are controlled by a small core of devoted citizen troops. In addition they field more traditional Mesopotamian troops.
-Siluhu barrack, slaves in all settlements
-Temple armoury, citizen troops with the taskmaster trait in cult centres
-Kassite military complex, regular Akkadian troops