Do birds move their babies to another location if they feel they are not safe?

Eggs cannot be moved to a different nest but in regards to chicks, it depends on the bird. Birds such as swans, geese, and ducks will sometimes ferry their young around on their back which not only keeps the chicks dry, it also keeps them safe. In regards to birds such as robins, sparrows, starlings, etc., the young cannot leave the nest until they have fledged so for those types of birds, the answer to your question is generally no, they do not until the chicks learn how to fly.