These long life pujas are very, very good ways for students to make offerings and requests to their Lama to stay. It is said that dakinis from other realms / places come to request the Lama to go to their place to teach. Sometimes, the Lama will begin to indicate and show signs that he will leave - he may express repeatedly that he wishes to go to another place, he may manifest sickness, start giving things away or advising pujas to be done. Students must be alert to their Lama showing any of these signs.
The dakini dance includes 5 monks who dress fully in very beautiful clothes made specially for the puja - they represent dakinis who come down during the puja and act to request the lama to come away. The puja master then approaches them and tells them firmly, "No, please leave, do not come here to request our lama to leave." There is a ritual of cutting scarves of the dakinis and sending them away. They leave the main prayer hall, one by one, as if they are expelled from the place.
At the same time, extensive offerings of new robes, 8 auspicious symbols, 7 royal emblems, new staff, Buddha statues (and other long life relics) are offered to the lama. It is a very good sign if he accepts them - an acknowledgment that he will stay longer. If the lama consents, you instantly see improvement in his health.
Anyway, whether a lama has controlled rebirth or not, if he starts to show illness, it is an excellent practice and offering on the part of students to offer him medicines, good food, serve him in any way that will help to alleviate his physical sickness. This is good for the student to collect merit, whether the lama has control over his death or not.