I remember reading a book, probably aimed at about a fourth grade level, in which there were aliens native to the moon. They decided to leave because more and more astronauts from earth were turning up. They headed off to live on another planet. Various semi-comedic adventures ensue. I honestly don't remember any details beyond that, though.
Matthew Looney series by Jerome Beatty jr. Essentially as you described.
The Matthew Looney books chronicle the adventures of a brother and sister, Matthew and Maria Looney, who live in the town of Crater Plato, on the Moon. In Beatty's stories, the inhabitants of the Moon are a fully developed non-human civilization. Beatty's fictional Moon inhabitants are an indigenous species, living on the Moon without the assistance of spacesuits, "breathing" vacuum instead of air. A recurring theme in the books is Matthew's desire to know more about outer space, especially the Earth. At the beginning of the series, he looks up in the sky at the Earth and wonders if anyone is living on it.