It's usually a long-drawn out process(I'm answering from an Indian perspective here-the Gupta and the Mughal empires).

Factors which catalyse the decline of the dynasties seem to include:

• Incursions by 'foreign' powers over a series of time and loss of vital frontier territories(the White Huns/Hepthalites in case of the Guptas,the inability to control cities that were part of trade routes on the frontier like Kandahar for the Mughals)
• In both cases,the people in charge of collecting revenue were incredibly powerful landlords/chieftains,who had their own armies/militias.It wasn't too difficult for regional people to assert them,often at the expense of the ruler at the center(the Mughal equivalent for this would be how the system of mansabdars---how they ceased to be transferred by the time of Aurangzeb and they assumed more and more power---a prime example of this would be Shivaji,whose father was a jagirdar,and was fighting to expand/maintain his jagirs with more autonomy in the initial stages of his guerrila warfare.
• Also,the Mughals had this in particular:intercine wars of succession between the princes which greatly weakened the imperial exchequer,and fragmented local loyalties across the empire.

Source: A History of the Guptas by RN Dandekar,The Princes of the Mughal Empire by Munis D Faruqi.

In Europe, dynasties could change when a family ran out of direct legitimate male descendants.

When this happened the family or "dynastic" name changed.

This was more likely in Europe, than in areas where polygamy was common among royal families.

In Chinese history, the dynasties generally last until they're overthrown in a rebellion or a war; in the 17th Century, global temperatures fell, so the Ming Dynasty's revenue dried up as a result of droughts and floods, and unpaid soldiers led by Li Zicheng revolted against the empire, capturing the capital until they were driven out by the conquering Qing Dynasty.

The Ming itself was established by a former beggar, Zhu Yuanzhang, who joined a band of peasants rebelling against the Mongol Yuan Dynasty, rose through the ranks, won battles against the Mongols, and became the most powerful warlord in China. He legitimized his rule by capturing Beijing and making ritual sacrifices to Heaven, thus claiming the Mandate of Heaven.