If I understand correctly the Aztecs subjugated cities through shock and awe, requiring only tribute to be given and not installing any governors.
If much of the dominion was simply held in fear of the Aztecs, would they have even been able to levy warriors from their conquered cities? Would they just have used armies from the Triple Alliance? How did they handle the logistics of that?
In Ross Hassig's book, Aztec warfare: Imperial expansion and political control (1988: 54-55), he mentions that the Aztecs would marshal troops in response to the severity of the thread. For small threats and small armies, seasoned troops were usually marshalled first beginning with soldiers from Tenochtitlan, then Tetzcoco, and then Tlacopan. Should a threat be serious enough or should the Aztecs require a greater number of people, they would begin to marshall troops from cities within the Basin of Mexico and then their surrounding territories and finally the most distant territories. If the threat was of the utmost severity or they needed their maximum amount of troops the Aztecs would marshall youths and old men to aid in the fighting.
Troops were marshalled by a high official such as a commanding general. The order for troops would work their way down the political hierarchy until it reached the calpolli level where neighborhood or barrio leaders would let those in their calpolli know of the order.
Most soldiers were only part-time soldiers. For the most part soldiers were simply farmers who were mobilized in the fall and winter to fight larger conflicts when grain stores were full and the wet season were over. The Aztecs did have professional soldiers such as the jaguar and eagle warrior orders and could, presumably, be called to fight during the spring and summer when the commoner soldiers were busy filling, planting, sowing, and harvesting their crops.
As for how the order would go from Tenochtitlan to the territories, Hassig does not say. My guess would be that a runner would be dispatched with the order to mobilize should the need arise for non-Triple Alliance cities and settlements to provide soldiers for the war effort.
Should territorial cities and settlements need to provide soldiers, Hassig (1988: 68) briefly mentions in the marching order that subject towns were obligated to provide soldiers to the Aztecs for war once a year, presumably as part of their tribute obligations. These tribute soldiers marched separately from the Triple Alliance cities, Tepanecs, Xilotepecas, Matlatzincas, and "the rest of the cities".
I hope this answers your question. Please let me know if you have any other questions.