I don't really know extensively about tenshugs or Long life pujas but I know there are several practices, especially associated with the deities of Amitayus, White Tara, 16 Arhats and so forth. These are practices to create the merit to extend the lives of ordinary practitioners, Geshes and so forth. These are very effective pujas and probably some of the most commonly sponsored in the monastery by lay people.
However, if its for a Lama who has control over their life and death, a very special form of tenshug called Kandro Tenshug needs to be performed. This puja invokes upon the Dakinis that have manifested to the Lama and they are requesting the Lama to leave their earthly body and to ascend to their realms in order teach and benefit.
Therefore, during the puja, 5 monks that are dressed in traditional Dakini costumes perform a ceremonial dance with damaru and bell, enters the prayer hall. They hold 5 colored khatas that are tied to the Lamas throne. Then, at a crucial moment in the puja, the khatas are severed, thereby requesting the dakini as symbolized by the 5 monks to return to their abodes and not call upon the Lama anymore. This is as far as I know about
Big uncle - what you say is correct.
The tenshug is a ritual where students make requests to the lama to live long. The lamas are here to benefit others and to turn the wheel of dharma. They can do it here with us, or in other realms of existence, such as the realm of the dakinis.
Although we cannot see them, it is said that dakinis often request the lama to leave our realm and go to their realm instead to give teachings. Particularly when the students of the lama misbehave, or when the karma of our realm to have a teacher with us is weakening, a tenshug is said to be effective to create the causes fr the lama to stay while symbolically "cutting" the connection with the dakinis.
Apart from the cutting of the khatas, what is notable about the tenshug is the extensiveness of the offerings made. The tenshug is about collecting the merit for the lama to stay, and each and every offering item is very symbolic - for instance, offering a teaching throne, offering food that the lama likes, and so on. And of course, the more offerings, the better.
Of course, however, a tenshug alone will not be enough for the lama to remain. I have heard my lama say that there is the outer tenshug, and then there is the inner tenshug. What is the inner tenshug? Transformation, merit accumulation, and pleasing the Lama. Only with the inner tenshug, does the outer tenshug have effect. Otherwise it is an empty ritual devoid of any effect.