I'm most interested in cities in the Han dynasty due to its rough contemporaneity with the Late Republic and Early Roman Empire, but would be also interested in how things worked in other dynasties.
In the Roman Empire, cities were run by provincial aristocrats who adopted Roman high culture to ingratiate themselves with the aristocracy at the City of Rome to gain their patronage to advantage themselves and their families in municipal politics. Despite the fact that these cities had significant autonomy to organize and run their internal affairs, they often came to imitate the collegiate oligarchic rule of the City of Rome as a result of these efforts. Were there similar imitative tendencies and patronage in Imperial China between cities at the center and at those at the periphery?
I have some knowledge of the Latter Han dynasty via the three kingdoms that followed, not the early Han dynasty.
People from the area could fill in the lower ranks of local administrations, officer of merit or reporting officer and the like. However, a Grand Administrator at the head of a commandery (usually based on a city), his chief assistant (or two for a military frontier province) was appointed from the capital and outside the area. The Inspector of one of 13 Province (then, after 184 in the last few years before the civil war, governors for frontier provinces) was also from outside the province.
This was to counter the power of the local families by bringing in independent figures from the outside though the ability to do so and the willingness from one gentry figure to challenge another varied. An official might gain notice if he was able to punish some of the more out of control retainers and bring families into line. Or he could have to flee for own safety and/or spark a revolt which would not be so good.
Keeping the powerful families (your local officers likely were from the power families or clients of them while at court the family may have people there) onside and perhaps not investigating too hard their armed retainers or the tax rolls could keep the peace and, if they were appointed to manage your home province, they would do likewise.
In terms of the frontier provinces, depending on the situation and to the official as to how much they pushed Han culture and how much they respected local culture. Some like the Governor of You (twice) Liu Yu was popular, one reason being he would wear the local clothing of fur and felt.
Others went for education and curbing the rituals like first Wei Li then Ci Chong who headed Guiyang. Governments and officials could sway on this, the Han was quite happy for faith healers to help during the epidemics of the 160s and '70s (that it also tweaked the nose of the gentry may not have been entirely out of thought) but after one such group led a major revolt (the Yellow Turban rebellion) in 184, there was more of a clampdown. Such figures would destroy local shrines and stop sacrifices, teach the Han customs on marriage and ritual, set up educational to instruct the local people, using the law of law to stop the local ways of marriage.
Via such measures, force of arms and (not always welcomed by the Han) migration south and west (they had problems with loss of population in the northernmost regions), the Han imposed its rule. The border regions like You, Wu, Yi were of the Han and would consider themselves as such. The Confucian culture, the rituals, the education was mostly shared and when civil war came, the idea was very much to conquer the rest of China to unify it rather than independence.
While they considers them of China and shared many (with "help") customs, those on the frontier had differences from the Central Plains. The western province of Yi always had it's education more towards soothsaying and rhapsodizes while during the time of civil war, it wasn't entirely friendly to outsiders coming to power, there was a very strong regional identity. The south had an interest in divination and their own mystics, the scholarly Jing warlord Liu Biao's south Jing officers had different interests to his northern ones and the southern families of Wu would ensure they were the local powers rather than the northern families that had fled south at the start of the war.
Rafe De Crespigny Generals of the South, also The Administration of the Later Han Empire
J Michael Farmer: The Talent of Shu Qiao Zhou and the Intellectual World of Early Medieval Sichuan
Andrew Chittick Life and Legacy of Liu Biao : Governor, Warlord and Imperial Pretender in Ancient China