The dynastic changes in China were not simply political changes, a peaceful passing of the mantle from one family to the next. They were events in a very profound way. When the Mongols conquered Song China, it was not a matter of two armies meeting on a field, one winning, and thus becoming emperor, rather it was a decades long, titanic struggle that ranged across southern China, and was followed by further decades resistance. One might imagine great hardship suffered by those who lived in the path of the armies, as well as hardship (and opportunity!) that went along with the general breakdown of civic order. Chinese historiography had sort of conventions of describing these periods of transition, and even if we don't necessarily accept that cannibals roamed the land it is pretty clear that the events reverberated throughout society.
One problem with answering the question in detail is that China is a very large, very diverse sort of place, and when you start getting more granular in your analysis you also arguably get less representative. That aside, there is a very good work on this very topic called Crimson Rain by William T Rowe, which traces the history of single place, Macheng County in Hubei, throughout its history, with the bulk dedicated to the late Ming and early Qing. To start with a big picture, the fall of the Ming was not precisely a "conquest"--starting in the 1630s a number of peasant rebellions began springing up, mostly in the south, and grew dramatically in the 1640s, particularly under two different leaders, Li Zicheng and Zhang Xianzhong. To make a very long story short, the Manchus only got past the wall because Wu Sangui, the commander of a key fortress, decided he preferred them to the peasant armies, and Beijing fell to Li Zicheng, not Nurhaci (likewise, it was primarily Wu's armies that defeated Li). The actual consolidation of China would not come until 1671, after decades of war and peace.
Macheng was uncomfortably close to the action during the peasant revolts, and Zhang Xianzhong actually marched his army through the county (albeit with minimal depredation, as a local notable had been kind to him in the past, so the story goes). But even when the fighting was not directly in the borders, the breakdown of the imperial structure impacted daily life. In articular, the local elite turned to fortifying themselves in the hills, and the pervading social tensions of the place broke out into open conflict, including outright rebellion by bondservants (essentially serfs). In particular, banditry (which should probably be seen here as a form of social conflict rather than simply rapaciousness) became commonplace, and individual bandits formed an impressive degree of military power. The larger conflicts engulfing the empire were reflected in local conflicts, which may have had local causes and roots but took the form they did due to the greater history of the empire. And of course some places (although not many) were even worse off than Macheng. When Zhang Xianzhong found his progress stymied in central China he marched to Sichuan, where he essentially committed a genocide against the Sichuanese.
EDIT: It also occurs to me that you might be thinking more long term. In Macheng's case, there was a great difference between the Ming and he Qing. During the Ming, the county was famous for being a fertile source of degree holding officials, during the Qing it was something of a backwater. Rowe speculates hat the violence of the transition and new social formation that emerged from it are at least somewhat to blame.
There is a saying in China that "the mountain is high, and the emperor far away" which basically means that the edicts of the imperial center only go so far in local affairs, and this is true enough. But this does not mean that local life was isolated from wider events in the empire. The history of a given place may be of that place, but that doesn't mean it does not get shaped and conditioned by wars and deaths and policies of the center.
This is a great question. One of the problems with answering it is that we don't have many accounts from the perspective of the peasantry during the Yuan-Ming transition, but there were plenty of observing literati who recorded the fall of the Yuan. However, I can say that yes, it was almost certain that the average peasant at least knew about the dynastic change. Much of the northern and central China suffered under poverty and violence as Yuan authority broke down and its ability to command the military forces necessary to put down rebellions dwindled. The violent transfer of power from Yuan to Ming profoundly affected the daily lives of peasants, particularly those living in central and northern China.
The best peasant's account we have is from the founder and first emperor of the Ming Dynasty, Zhu Yuanzhang, who was born into a peasant's family. Of course, given that he became the first emperor of the Ming, the surviving records of his early life are naturally skewed to depict the Yuan as a failing empire and its Mongol rulers as inept, thus paving a historiography that legitimized his rebellion and rule. Zhu Yuanzhang was born in modern Anhui, and his family was extraordinarily destitute. During his youth, Anhui was the center of a major religious rebel movement, the Red Turban Rebellion, which according to records drew most of its followers from similarly destitute peasants. Disease killed most of his immediate family, and he lived alternatively as a monk and a beggar until he joined the Red Turbans. The story of how he became the most powerful military commander of the Red Turban Rebellion isn't pertinent here, but I can offer a description later if you'd like. The point is that by the 1350s, epidemic, starvation, and poverty had struck central and northern China to such an extent that rebellious movements like the Red Turbans grew by enlisting disaffected peasants.
Outlaw groups and sectarian movements like the Red Turbans sprouted frequently in the last decades of the Yuan, often effectively becoming military governors who displaced the power of Yuan-appointed officials. The Yuan government occasionally attempted to co-opt these groups by granting them titles and an air legitimacy, but in actuality the power of the Yuan court over them was virtually nonexistent in the 1350s and 60s. Religious sectarians, mostly Buddhist, formed organizations that actively worked against the Yuan state and often sparked rebellions, including the Red Turban Rebellion that became the operational and military foundation for Zhu Yuanzhang's Ming Dynasty. On a smaller scale, as the Yuan court's control over its Chinese armies (predominantly stationed in the south) slipped, small-scale outlaw and bandit groups became a common nuisance/terror for rural regions. The average peasant would have probably noticed changes in who governed them, would have absolutely noticed if a lack of proper government allowed banditry to flourish, and would have probably noticed changing patterns of extraction as Yuan-era regulations regarding tax rules and limitations lost meaning under "outlaw" governors. Below is an account of salt-field laborers organizing to combat organized criminals who were harassing them:
Hsiao Ching-mao was a man of Ko-chou village in Lung-hsi district of Chang-chou prefecture in Fujian. ... During the Chang-chou uprising of the later Chih-yuan reign period [1335-40], Hsiao led the people in his locality to construct fences and barriers for their protection. ... Yet it happened that someone from a neighboring village guided the bandits through by a devious route. Hsiao Ching-mao was taken captive. The bandits ordered him to bow to them. He responded: "You are bandits. How could I bow to you?" The bandits' intent was to force him to surrender in order to impress the people. Hsiao cursed them. "Traitorous bandits, what wrong of the state could justify your treachery? Your clans, your villages, what have they done to deserve the evil consequences of your acts?" The bandits said to each other, "We have killed many of the [Yuan] government's army officers and civil officials. Brought to our strongholds, they have all cowered in fear and begged for their lives. Never have we seen any as stubborn and unbending as this tough old devil." (1)
This account offers an interesting alternative to Zhu Yuanzhang's narrative, because it suggests that Hsiao did not hold animosity towards the Yuan state or its Mongol overlords. However, this account is usually dated to 1338, which was just before major peasant revolts started to frequently grow into anti-Yuan rebellions. It is likely that many peasants, particularly in the south, did not actively work against the Yuan Dynasty or wish its collapse. At the risk of dipping into generalizations, my professional assumption is usually that most subsistence farmers didn't care too much about who sat at the top of their polity's political structure, as long as the farmer did not have to deal with stressors like frequent violence and the spectre of starvation.
To to summarize, whether or not a given peasant cared about the Yuan-Ming changeover probably depended greatly on where he/she lived. Southern China's economy probably grew under the Yuan era, so southern peasants may have been less inclined to hold a strong opinion beyond maintaining their livelihoods.
Sources:
(1) Tao Zongyi, Chuo Geng Lu 輟耕錄 (1366), preface; also in Yuan shi 元史 193, p. 4388, translation provided in Mote and Twitchett, below.
Timothy Brook, The Troubled Empire: China in the Yuan and Ming Dynasties (Cambridge, Harvard University Press, 2013).
Frederick Mote and Denis Twitchett, The Cambridge History of China, Volume 7: The Ming Dynasty, Part 1 (New York: Cambridge University Press, 1988), "The Rise of the Ming Dynasty," pp. 11-57.
EDIT: typo corrections, minor elaboration, formatting corrections
Unfortunately we do not have access to a wide variety of primary sources from peasants in imperial China. The best sources we have are county gazetteers (地方志）which are incredibly useful and frankly fascinating. This can be largely attributed to a very low literacy rate in China historically. The time and effort it took to learn the written language was substantial and expensive. Moreover, the written language, classical Chinese, bore little resemblance to the spoken language.
To some extent, the dynasties were somewhat interchangeable in large part because they all utilized similar logic to justify their rule. The Manchu's for example often sought to portray themselves as the rightful heirs of the Ming dynasty's artistic and literary traditions. You can see this in the paintings and other works of art collected by the Qing Emperors, especially the Qianlong.
I suggest taking a look at "Spring's Peaceful Message" by the Italian Jesuit painter Giuseppe Castiglione (Lang Shining) commissioned by the Qianlong.
What we see here is an effort by the Qianlong to portray himself as Chinese and a participant in the filial traditions of Chinese rule. Of course the Qianlong was not Chinese, he was Manchu. Towards other audiences he (an other Qing/Yuan emperors) attempted to portray themselves as very Chinese whereas they attempted to portray themselves as Manchu, Mongolian, and Tibetan towards other audiences. Note the hairstyle, both figures in the painting are not wearing the Manchu que but rather a traditional "Chinese" literati hairstyle. Without getting into the cosmic and spatial connotations of the painting, I think this piece reflects the "universal" nature of Chinese dynastic rule. The language used, religious and moral motifs, and logic used to rule was surprisingly consistent for large stretches of imperial history.
Of course, as another response notes, the change of dynasties was often accompanied by dramatic political and social upheaval and the imperial period was by no means entirely uniform. But for the most part, to the "Chinese" audience it would in many ways appear the same largely because of efforts to maintain a sense of continuity between rulers.
I'm primarily an expert on the Qing period so I'm sorry I can't provide information more relevant to the Yuan and Ming periods but I think these sources may prove useful in understanding these themes:
Wu Hong, “Emperor’s Masquerade: ‘Costume Portraits’ of Yongzheng and Qianlong.”
“Like a Cloudless Sky” & “When Words Collide,” in Empire of Emptiness: Buddhist Art and Political Authority in Qing China