Bluebird is a great game for young children. I have found it effective to have the kids press hands together rather than hold hands in order to make the windows. The person who is "it" flys around through the windows and lands in front of someone at the end of the song and asks, "(name), would you like to be the bird?" If the answer is "yes" then the new one is "it" and the previous one takes their place. This game can be played with multiple birds or students can choose what kind of bird to be in which case there might be two questions, "Would you like to be a bird?" and "What kind of bird would you like to be?" As the teacher, I like to eventually drop out of the singing and let the students sustain the singing game. Whoever begins the song, of course, chooses the starting pitch. I don't think the starter needs to give the pitch beforehand either. We start and everyone settles in on a pitch. The idea is to keep it complex and interesting.
Once the children know the song and game, it can be partnered with Skip to My Lou. In fact, there are quite a few songs that will partner with these.