History

are there examples of a declining empire that the turned the trend around durably? 

As in: things are going downhill, someone comes and turns it around for the better

Investing at cherry.xyz | writing at luc.cx

Haven't all long-running empires gone through several cycles of that? Rome (several instances of near-death in the BCs), China, etc

True, but often the empire is transformed so radically that it is a new empire in all but name: Rome redefining its citizenship after the 3rd-century crisis, the Russian Empire with its political restructuring after the ‘Time of Troubles’, Britain swiftly decolonising post WWII, to give a few examples