They would most definitely claim it as it was happening. One good example is of the Song. In the 900s as the Song was seeking legitimacy, a history of the Five Dynasties was published, written primarily by Xuē Jūzhèng 薛居正, arguing the case that the mandate of heaven was there, and that it now belonged to the Song. This is perhaps the most important motivation behind the writing of the History of the Five Dynasties; it legitimised the Song as the new single power. It wasn't the only time this was done with official histories by any means.

This was initially somewhat problematic since it meant having to justify that any of the five dynasties also had the Mandate on their side from which it could then fall on the Song.

Still, the concept was very much something people were aware of at the time of power shifts.

One of my fellow Asia flairs will certainly have something to add. /u/Jasfss and /u/FraudianSlip come to mind.

One contemporary I can think of is Zhang Jue, leader of the "Yellow Turban Rebellion" which helped precipitate the warlordism which ended the Han Dynasty. A string of later Eastern Han emperors in the 2nd century CE were child emperors. Han Chongdi, one of them, didn't even make it to three years old. The adult Emperors Huan and Ling, who followed them, attributed this to control of the court by Emperors' mothers, the Dowagers, and their family networks. As often happened in Chinese history, the Emperors placed their trust and authority in the eunuchs in their private chambers. These eunuchs, the "Ten Regular Attendants," ruled with extreme favoritism and corruption and this led to widespread discontent.

Zhang Jue, aka Zhang Jiao, attributed a string of natural disasters, high taxation, corruption, and myths like a dragon appearing in a well in Mopo to the Han's loss of the Mandate of Heaven. The Han dynasty's red fire sign waned as the Yellow sign of Earth, which Zhang claimed in his Taoist association to "Huangdi," the Yellow Emperor, waxed. Much of the momentum of the Taoist Millenialist movements of Zhang Daoling and later the Five Pecks Sect of Zhang Lu was capitalized upon by Zhang Jue, who styled himself the "General of Heaven" and his movement the "Way of Peace."

If this was so, Heaven, or "tian," was fickle or misleading. A series of warlords rose up to help the weak government vs. the massive but poorly armed and poorly led peasant rebellion. They eventually quelled the rebellion, Zhang Jue died and his brothers were killed, but central power was fractured and eventually the "Three Kingdoms" were born from the chaos, briefly united into the Jin Dynasty before the Revolt of the Nine Princes and northern horse confederation invasions once again divided China into disparate dynasties for three hundred years.

Best source is: Chen Shou, Records of the Three Kingdoms, Yunnan People's Publishing House, 2011.

I think it's worth noting just how superstitious the general public was, even up until late Qing dynasty. Throughout China's history, playing the Mandate of Heaven card was one of the few ways of influencing a group of people to revolt (religion, which is a tad bit relevant, and swearing blood-oaths were some of the other ways.) As /u/JiangZiya mentioned, Zhang Jiao was a prime example of successfully claiming a loss of the Mandate of Heaven to gain momentum in his rebellion. Even before him, Chen Shen & Wu Guang's Dazexiang Uprising (another eventually failed uprising, against the Qin Emperor) began with them planting white cloths that said 'Lord Chen Shen' on it into the abdomens of fish for people to find. This (along with some other rumors that the two spread) caused people to believe that Chen Shen was destined and appointed by the Heaven to overthrow the Qin Emperor.

In every corner of China's history of conflict, you'll find some ambitious rebellion starter using the Mandate of Heaven as an excuse to corral the people around them and gather an army. You'll find generals using it to boaster the army's morale. You'll find those trapped in a dire situation actually questioning themselves whether it was the Mandate of Heaven that they should die. And of course you'll find the winner of the conflict always having the Mandate of Heaven, because the winner writes the history books.