Well it is a lullaby-in Aajfe's version, so it's sung to children. I always thought that the song was similar to the idea I named above, allowing your child to pretend they are going somewhere else in their imagination. Maybe the mother is trying to get her children to pretend they are birds traveling to far off lands in their dreams.

It was, supposedly originally intended as a marching song and so that would most definitely be the case if you were marching somewhere. (Although I don't know how far these lyrics date back)

Who in their right mind would sing songs directly to a bird?

Of course, again none of this is documented and I too would like to see another version collected of the song. Some type of date would be excellent. It could very well be a song about birds or sung to a little bird.