I haven’t done anything like this before, but it reminds me of how I did a “snake” game a long time ago. 
My first guess would be to make the head control the movement of the dragon. With each step of movement of the head, you take the direction vector of the head and ease the direction of the next bone to the head (bone 2 → head bone) until it matches the head direction. Then you’d repeat that (bone 3 → bone 4 , bone 5 → bone 4), with each bone following the vector of the one before it.
The way I’m imagining it is a chain on the ground. If you grab one end and pull it in one direction, eventually each link will go in that direction.
And beyond that I imagine you could mix in some kinda vertical sin-wave movement to make it less “snake” like. My 2 cents!