Okay this one is weird. There is this anime I love called "The legend of the galactic heroes" SPOILERS near the end the Alliance of Planets is conquered by the Galactic Empire and there are several scenes of the conquerors trying to establish order but low level public servants refuse over and over to hand over documents, make decrees, or in general recognize the conquering government as legitimate.
That sequence ends with one of the main characters saying that if those people had been the ones leading this nation, and not the corrupt politicians at the top, then probably they would not have been conquered.
Then I heard somewhere that the writer of the show pretty much stole this from Chinese history. I remember reading somewhere that this is exactly what public servants of a previous dynasty would do when their government fell, and that this was seen as heroic in a sense.
I know this is super vague, but if it is true, I want to know more about this trope and I want to know about individual examples if we know about them
From the fall of the Han and three kingdoms perspective, no. Or at least, not that far. There were the odd figures here and there who chose retirement and were usually treated with respect but if you refused to serve the new administration, you retired. Or led a military revolt but I get the impression that sort of act of loyalty (ditto suicide in protest) is not what you are looking for.
The period was between those who recognized each other as part of the same country, even if there were strong regional tensions. It may have happened in other eras but I can only speak for this era I'm afraid.
One example of the sort of thing that went on:
In 205, the Yuan clans rule over the north was ending as Yuan Shao's children had, thanks to an incompetent heir situation, turned on each other. Losing the support of their officers and allowing the powerful warlord Cao Cao to take their lands. The eldest Yuan brother Yuan Tan had died, their capital of Ye had fallen and now the brothers were forced out of You province and to the Wuhuan by Jiao Chu.
Jiao Chu sought to confirm allegiance to Cao Cao by a blood oath of all the officials that he could muster (You had not been fully under Yuan control and different leaders took different approaches) but his attendant officer Han Heng refused. "I received Excellency Yuán father and sons’ generous grace, now they are broken and destroyed, my wisdom cannot save them, my valor cannot die for them, and my righteousness is already deficient. If then you turn to face north to the Cáo, that I cannot do.”
This was awkward as Jiao Chu had threatened death to any who refused to swear allegiance to Cao Cao (and by extension, himself as the new Inspector of You). Jiao Chu thought it wise not to press the issue and start regime change by such a killing but let Han Heng go free, a position endorsed by Cao Cao. Han Heng fades from history, Jiao Chu was probably killed by Yuan loyalists revolt.
In such occasions, it preserved a bond between patron and client, a sense of honour and perhaps looked ground on the ground of the local area which was sometimes the main concerns of gentry officials. Or, for personal reasons, they might not wish to serve but overall, they didn't question the legitimacy of the new masters. The rulers meanwhile tended to accept this, recognizing loyalty to old master could transfer to themselves and that it might not look so good if you killed men for being honourable.
I will use the downfall of the four dynasties in order (Han, Shu, Wei, Wu, I don't have the sources for Jin) and any post-fall objections. I will also mention the novel Romance of the Three Kingdoms (and make clear when using the novel) because the novel has become people's perception of history and such things may have shaped any such ideas.
The Han collapsed in 190 into civil war with the child Emperor Xian bouncing around from "protector" to "protector", including engineering his own escape from Li Jue's military junta, till 196 when Cao Cao arrived and took control. Relations between subordinate and the Emperor were not easy, there were outbursts, assassination plots met with killings of the Empress, concubine, some of Xian's children and inlaws. Over time, Emperor Xian's friends were killed or removed from the court. In 15th May 220, Cao Cao died but was replaced as King and warlord by his literary critic son Cao Pi, in December Emperor Xian abdicated and the Wei dynasty came into being.
So what shows of defiance were there? In 213, Cao Cao had put three daughters into the Emperor's harem, two years later Empress Fu Shou was accused of plotting against Cao Cao and killed along with her children, one of the three Cao daughters Jie became Empress. As Pi's sister and Xian's husband, facing demotion from Empress, she chose her husband. She refused repeatedly to hand over her seal of office, rebuked the envoy and threw the seal over the railing, proclaiming "Heaven will never bless you!", every attendant discovered they liked looking at the floor.
Then there was Yang Biao, a man of great family and a noted scholar who had worked in Han history projects then had served the Han in the very highest ranks till Cao Cao came along. Tortured due to his wife being sister to the false Emperor Yuan Shu but, since no reasonable charges could be brought, he was let go and when sacked in 205, retired on grounds of illness with leg cramps. His talented son Yang Xiu would serve on Cao Cao's council till executed for his giving council secrets to potential heir Cao Zhi with Cao Cao not pleased with Yang Biao showing sorrow. Cao Pi was an admirer and wanted Yang Biao as Grand Commandant of the new regime but Yang Biao refused on the grounds he had failed as a Han Minister. Was given special honours but remained in retirement as he worked on the histories.
In the novel Romance of the Three Kingdoms, Cao Jie is shocked but her protest is given to a fictional figure called Zu Bi, keeper of the seal, who has to be beheaded before the armed generals Cao Hong and Cao Xiu can take the seal. Emperor Xian protests loudly but is threatened into doing it. People feel sorry as the former Emperor leaves and Heaven sends a signal via wind that makes Cao Pi sick for days.
In 263, corruption at court and the military aggression of fame-seeking chief general Jiang Wei had taken its toll on the state and the court (or the chief general who had survived a court plot to remove him) were not popular. Sima Zhao, controller of Wei, sent three armies under Deng Ai, Zhong Hui and Zhuge Xu to attack, Jiang Wei's new defences failed at Hanzhong and Deng Ai's surprise advance through the mountains saw the capital forces destroyed under Zhuge Zhan and Chengdu was no longer holdable. Emperor Liu Shan was persuaded by soothsayer Qiao Zhou to surrender, playing on Liu Shan's kindness and arguing other plans like flight to other areas or powers were hopeless.
Generals like Lou Xian switched to the conquerers once they were sure of Liu Shan's treatment while Jiang Wei (whose army is said to have been upset at surrender) tried to use the divides between the conquering generals and between Sima Zhao's court to engineer a recovery but he and Zhong Hui's coup collapsed quickly as soldiers refused to support it but the rioting in the aftermath had a cost, including Liu Shan's eldest son Liu Xuan dying.
Some of Liu Shan's family took exception to the (controversial) surrender. Son Liu Chen had advocated a final stand as befitting an Emperor, Shan refused so Chen killed his family and himself in protest at the dishonour (the novel makes Chen's wife an active participant). According to the pro-Shu historian Xi Zuochi, one of Shan's concubines Li objected to the palace woman like herself being given to unmarried Wei officers, she could not take a second or third humiliation and took her own life.
The officers of Shu were given marquis and posts to win over the local notables, the only one I can think of who refused was Xi Zheng who chose to leave his family and act as Liu Shan's adviser at court. When Xi Zheng became mortally ill and returned home, the Jin court praised him for his loyalty and made him Administrator of Baxi.
The novel goes for many unnamed generals dying with Jiang Wei, two old generals Dong Jue and Liao Hua dying of grief, highlights the "heroics" of Jiang Wei and Liu Chen while lambasting Liu Shan and Qiao Zhou.