During The Shogunates
Usually the establishment of Kamakura Bakufu marks the beginning of Kamakura period. Yet this did not immediately end the Insei system. Though Kamakura Bakufu took over the police force and ruled Eastern Japan, the authority of Emperor and retired Emperors remained. The court and shogunate coexisted till the end of Edo period. At least at the early Kamakura period, Chiten kept substantial power over many important decisions.
However, when Go-Toba, a grandson of Go-Shirakawa and Chiten at the time, planned to overthrow Kamakura Bakufu and failed (Jōkyū War), the power of the court, particularly that of retired Emperors was markedly cut down by the shogunate.
Even after the Jōkyū War, the cloistered rule system continued to exist, at least formally, for two centuries. There were movements to take the authority back into the hands of Emperor at the throne, such as the Kemmu restoration by Emperor Go-Daigo, but in general a retired emperor presided as the head of the Kyoto court, with the approval of the Bakufu.
There were a few examples of retired Emperors supervising their successor later in the Edo period. The last to bear the title Daijō Hōō was Emperor Reigen in 1686.
Read more about this topic: Cloistered Rule