A further note. The Han Shu clearly mentioned the Alani as been subordinate to the Kangju (属康居), and the Alans have raided as far as Armenia and Parthia in the late 2nd century BC, so technically, the Kangju's influence extended into the Caspian and even the Black sea. The Han established diplomatic relations with the Alans under Wudi, but it didn't seem to have been regular, so there was only indirect tributary through the Kangju (The Kangju likewise nominally submitted to the Xiongnu, so that's probably why you see some Turkish made maps showing Xiongnu empire going to the Caspian as well, rivaling the later Gokturk Empire in size); that's probably where the Han influence up to the Caspian idea came from.That depends on the map. I've heard of claims that Gan Ying's soldiers conquered all the way to Caspian, another one of those nonsensical wikipedia facts that cannot be verified in any primary source. Gan Ying merely brought an embassy, not an army (and wikipedia often gives the figure of 70,000, which is again completely baseless). The Kangju seem to have reached the Aral, there is too little sources on whether they reached the Caspian however, we know that to their northwest were the Alani. Han influence over Kangju is probably in the same manner as the Persian influence over the Sacæ nomads; that is nominal submission and token tribute (I am not even aware the Sacae had hostages at the Persian court, while the Kangju did have rolyalties at the Han court). Han influence over the Xiongnu was much more significant than the Kangju. Hu Hanye Chanyu visited the Han court personally in 51 BC, and again after 36 BC, later Xiongnu rulers were required to bow and even kowtow to the Han envoy. By Wang Mang's rule, the Han tried to meddle directly in Xiongnu internal affairs. First, he forbid the Xiongnu from receiving Wusun, Wuhuan, and Western region refugees, then he forbid the Wuhuan from paying tribute to the Xiongnu, then Wang Mang forbid Xiongnu rulers to have two character Chinese names. After 10 AD, he appointed all 15 sons of Hu Hanye Chanyu, in an attempt to split the Xiongnu into 15 polities. Wang Mang then send an envoy to the borders of Yunzhong county and managed to get two of Hu Hanye's sons (one of which was a lord of a fief) to come to the Han court, forcibly gave them the title Chanyu and made them stay in Changan. But by this time, the Xiongnu ruler has had enough with the New regime and decided to raid the frontiers on a large scale.
Han control over the Wusun is very real however. In 53 BC, the Han stationed some 3,000 soldiers near the Wusun winter capital Red Valley (Chigu) (southeast of Issy Kul, around the area of Suyab). The Protector General even petitioned the Han court to abolish the Greater Kunmi ruler (grandson of Han princess Jieyou and Wengui mi) because he was too weak of a figure but was turned down by the emperor. Not only did the Han have the ability to abolish the Wusun ruler, even prominent Wusun officials carried Han seals and had to have Han recognition for succession. In 5 BC, the Wusun smaller Kunmi raided the Xiongnu but was defeated and the Xiongnu wanted the Wusun to provide a prince as hostage. The Han then send an ambassador and told the Xiongnu that both the Wusun and Xiongnu were Han subjects so it is inappropriate for the Xiongnu to have the hostage of another Han subject. The Xiongnu reluctantly gave back the hostage to the Wusun. In 1 BC both the Xiongnu ruler and the Wusun ruler (greater Kunmi) visited the Han court in person.