Pretty lax. Sometimes the Chinese held hegemony, sometimes the Xiongnu, sometimes neither throughout the Han period. Small states in the region tried the delicate balancing act of making concessions to both powers in order to try and preserve their own independence. For instance Loulan in 108 B.C and 92 B.C sent one prince each to the Hand and Xiongnu courts as hostages. Eventually the Chinese managed to gain control of the region after a successful campaign against Turfan in 90 B.C. We should note that the army they used was drawn from six different states of the western regions, showing Chinese influence. Some agricultural colonies were set up, walls extended and Chinese rule imposed. In return for tribute (gong) local rulers were presented with Chinese titles, normally a marquisate, along with an official seal of office which entailed them carrying out administrative duties on behalf of the state. These were stripped from rulers who did not do as thy were told, as happened when the king of an area of Turfan was judged to have been negligent in his duties in 153 A.D. He was stripped of his title, which was granted to a neighboring noble. When the Tang extended their power into central asia long after the fall of the Han these regional nobles still retained their Chinese titles and seals long after the court which had granted them had faded away. I don't believe that there was any overt sinoization of the region so to speak, their control was too fragmentary.

Mark Edward Lewis "The Early Chinese Empires: Qin and Han"