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Continuity, in its very essence, is the (destructive) transfer of a consciousness from one body to another, with the mind-altering properties of Kuva playing a key role in it.
The ritual consisted of Orokin whose current bodies have grown old and frail, going to a Yuvan Theatre, where selected young and beautiful potential hosts would be paraded for display and selection. The Orokin would then choose the person (Yuvan) whose body they would use as their new body, after which the Orokin would invade said person's mind under the influence of the liquid Kuva until only the Orokin's mind remained inside the body, erasing the former host's consciousness.
It also is suggested by Cephalon fragment recordings and the younger Grineer Queen's dialogue that Orokin can also participate in Continuity even after their body has died. Whether or not this is theoretically possible with any person or only those whose mind has been affected by Kuva once before remains currently unknown.
Because of the nature of Continuity, it might be so that we will see (more) Orokin returning in the future, as Continuity is possible even after death. This mainly might be the reason why the Grineer under the leadership of Worm are seen harvesting Kuva, to prepare for the continuity to, if the player selected moon or neutral choices in the War Within, resurrect the fallen queen. It's also up for speculation how the ritual works in detail.