My knowledge is of the Han and three kingdoms eras. Which didn't have the Admonishment Branch but may be able to give you an idea of what the Censorate did.

The Censorate, nominally under the Minister Steward and headed by Palace Assistant Imperial Clerk who led Imperials Clerks (senior ones known as Preparer of Documents) who were able to investigate and report on crimes within the court, sometimes appointed as envoys or special investigators when an Emperor required it. They read reports from the provinces and all memorials to see if anything was wrong, sometimes passing it up the like to the Secretariat. They could also supervise the Imperial library and act as a staging post for rising scholars. They could also be sent to Kingdoms to supervise ceremonies involving Kings or to lead troops.

Below the Imperial Clerks were the Internuncios (supposedly noted for strong voices) who took part in ceremonies and could act as envoys or to supervise projects in the Empire, they could take up military command when required with the training camp at Liyang often under one such figure.

The Admonishment Branch was not around at this time and at the time the Censorate were meant to be agents of the Emperor against a bureaucracy he couldn't always trust. Though they could sometimes fall into the hands of factions.

All sorts of people could criticize an Emperor. Senior ministers, members of the court including counsellors, friends, people who had been offered a job and used it to lay out why they were rejecting it, local officials could use an omen or another reason for reporting to criticize if they dared.

Subjects could range from foreign policy, domestic policy, need to restore rituals, choice of Empress or heir situation, choice of political allies (i.e eunuchs or anyone not of the old Confucian class was wrong). Drunkenness wasn't the more common charges in personal life but too much hunting (or too keen on taking on tigers at close range), lack of restraint in spending or behaviour, need to be more filial to the Dowager, too much travel, too little time at work, too little time in the harem in one case.

The tone of criticism varied depending on the person sending in the memorial, the power dynamics involved and sometimes the Emperor themselves. Some could be gentle and discreet advice, some could be firm warnings or stern rebukes, in a few cases, the complainer could be having an absolute go at the Emperor in strong terms.

If the Emperor (or the controller) was absolutely determined to kill you for that criticism? How fast can you leg it to exile and how good are you at hiding? If the memorial crossed a line the Emperor deemed unacceptable (be it the content, or the tone) or simply the incident reminded them they didn't like you due to something in past (failure to give them a favour, past slight, being witter on whose bum-hair was hairier) and now you have provided an excuse: Your doomed.

There were political costs for such executions, an Emperor and his bureaucracy did not always see eye to eye as to what was a justified execution. Executing a man for doing his duty and offering honest advice (as it might be painted) to the throne looked intolerant, that you couldn't handle men of true talent and virtue. There might be protests from ministers and risk a rupture with them, some might refuse to serve such an intolerant regime, an Emperor's reputation could be damaged by executions that were deemed unfair. This could stop the ruler from going as far as he might want, he had to make a calculation.

In the three kingdoms, there was an incident of one talented but troublesome figure called Mi/Ni Heng who ended up being the parcel in pass the parcel between warlords Cao Cao, Liu Biao and general Huang Zu to avoid blame for killing a noted talent. It also wasn't unknown for an Emperor to reign in officials who were outraged on his behalf at the criticism.

How much officials were not Emperor (or like an Emperor) could get revenge? If they could get the Emperor's ear and persuade him to act or if that isn't going to happen, they might need enough faction strength or connections to get a troublesome figure moved to a nice sounding post that gives them less authority. If that isn't an option then if connections within the law officials to force a legal case through, they might be able to get revenge within the law but others might seek to block or protest such an effort and other officials might not be too pleased, an official would be using political capital to do this.

The moving around seems more like local magistrates or law officials whose taking on the local powers might sometimes mean moving them on to spare their lives and defuse tensions. The Censorate themselves would not be in the interest of the Emperor to move around. In terms of popular tales, there was a saying about imperial clerk Huan Dian, rider of a piebald horse and known for his honesty, "Walking, walking, better watch your step. Beware the Clerk on the piebald horse."

Yes, they were of the scholar-gentry branch.

Sources:

Rafe De Crespigny, Administration of the Later Han and Inspection and Surveillance Officials under the two Han dynasties

Annals of Wei and other SGZ by Chen Shou, annotated by Pei Songzhi, translated by Yang Zhengyuan

ZZTJ by Sima Guang, translated by Achilles Fang