Not sure about the current title so suggestions would be appreciated.
A sapient species is in serious trouble. So serious on fact that they're in very real danger of going extinct and/or having their planet destroyed.
Fortunately for them an alien species has noticed their plight and comes to their aid, saving as many members of the species in question as they can, often by relocating them to another planet.
- Arthur C Clark 's short story Rescue Party involves a ship full of aliens coming to Earth shortly before the Sun goes nova with their mission being to try to save as many people and as much of the culture as possible. Subverted in that when they arrive they discover that humanity has managed to save themselves.
- Isaac Asimov's story "Blind Alley" features a race the humanity saved. The problem is, they now refuse to procreate, seeing no purpose in existence now that they know the Galaxy is ruled by humanity.
- Another story by Asimov, "The Gentle Vultures", features a variation. A race knows all other races will one day to destroy themselves in a war. Since they have no way of preventing that, they let the war come, then rescue the survivors and breed the violence out of them.
- Featured in the Star Trek: The Original Series episode "The Empath". A sun is about to go supernova and wipe out all life in the solar system. The Vians can only save one of the races living there, so they test members of the races to determine which is most worthy. At the end of the episode they choose to save the race of the female character who appeared in the episode.
- The basic plot of Star Trek VI: The Undiscovered Country was that a Klingon moon had blown up and Star Fleet had to negotiate with them in order to mount an evacuation of their homeworld.
- In Star Trek The Next Generation the Enterprise crew save a planet from destruction after Data comes in radio contact with a small child. However it's against the prime directive so directly contact the people so they save the planet without informing the inhabitants.
- There was another Star Trek TNG episode where they transplanted the inhabitants of a planet right before a catastrophe rendered it uninhabitable. Actually, Worf's brother Nicholas beamed them into the holodeck without permission, forcing the crew to find a suitable planet for them. Picard is outraged, but goes along. Nicholas, it turns out had a very personal reason for saving them, one of the women is carrying his child.
- In the backstory for Mass Effect, Rakhana, the drell homeworld, suffered an ecological collapse that, as they lacked interstellar flight capacity at the time, would've wiped out their species if then hanar hadn't found them and mounted a large scale rescue operation, transporting hundreds of thousands of drell to their own homeworld, Kahje.
- The Alternate History timeline/Mass Effect fanfiction On the Shoulders of Giants has an interesting variation with the quetzal and humanity. After finding three hundred and twenty-six quetzal in suspended animation on Titan, humanity launched a mission to rescue them and bring them back to Earth (or, more accurately, the Moon). It is later discovered that those quetzal are all that remains of the species. Notable in that while this trope usually involves a more advanced species rescuing a less advanced one, on this occassion it was the other way around, with humanity not even being capable of FTL travel at the time.
- The first time the Animorphs meet the Ellimist, he proposes to do this, by taking them and their families elsewhere to save them from the Yeerk's inevitable defeat. In order to convince them of this, he transports them into the future, where the Yeerks have won, Rachel is a Controller, and the formerly underground Yeerk pool is now open-air. When they get back, they agree to the Ellimist's deal... and nothing happens. Rachel realized the point of this was actually to show them the location of the Kandrona ray generator, which the Yeerks need to survive, and is no longer hidden in the Bad Future.
- There was a Scrooge McDuck story where two planets' orbits took them within less than a mile of each other, with a sentient species in danger of extinction on one of them. He tethered one planet to another and instructed the aliens to climb to safety.
- In Jack Mc Devitt's Omega (part of his Priscilla Hutchins series), humanity finds a species at about a Renaissance-level of development, directly in the path of the civilization-destroying interstellar artifacts known as "Omega Clouds". They don't want to disrupt the civilization by making contact, if they can avoid it, so they devise an elaborate plan to help the Goompahs save themselves.
- Doctor Who: That's what it turns out the star whale was doing for humanity itself in The Beast Below.
- The aliens/angels from Knowing save a selection of children to restart humanity on another planet after a massive solar-flare completely destroys Earth. How altruistic they really are however is up to debate.
- One story in Buck Godot Zap Gun For Hire had Buck trying to obtain the assistance of a Sufficiently Advanced Alien to help rescue the population of a planet whose sun is about to go kablooey, because human technology is insufficient to get them all off the planet in time. After he manages to explain the situation to it, it casually remarks that it would be much simpler to leave the people where they are and move the whole planet to another solar system -- which it then does.