The Yuan Dynasty assigned official posts in the Uigher 畏兀兒 area , like 提刑按察司，都護府 and in the western part 宣慰使元帥府，都元師府 units. In 1308 AD households registry of Samarqand regions was presented to Yuen court. Based on that Yuen collected tax from Samarqannd, 塔刺思城 etc. Tax collectors were sent by Yuan Dynasty.interesting.
how do they collect taxes? the basement to collect tax is the existence of some documents, on the population, on land, on mines...
and you are sure the "within" yuan events were not happened during or before the war?
does chagatai or ilkhanate had any of those doc, beyond what a map tell us where they are?
how do they enter the pass? do they enter by claiming collecting tax, or they enter by claiming visiting yuan emperor? do they even need a diplomatic document to enter the pass?
From Yuan Shi 元史 volume 22 ,Wu Zhong Record 1 武宗紀一
After that the eastern part Qara-hoja 哈刺 火州 came under direct rule of Yuan as 總管府 unit. Other areas were under rule of Chagatai royalties.
Han immigrants were located in Bes Baliq 別失八里，Almalik 阿力麻里 etc. Worked as farmers, cotton weavers and garrisoned soldiers. About 30,000 Uighur artisans were absorbed to work in Dadu (Beijing).In the capital city of Chagatai Khanate Almalik, Hans, Mongols and natives lived together.
There were Yuan garrisons in Bes Baliq, 闍鄽，幹端 etc.
If I remember correctly the Yuan bereaucract issued passes to travellers in silver plates to allow travel from khanate to khanate with no trouble.