The baby dragon is just following me as I mine into the mountain (I'm mining fast because I have a dev-cheat pickaxe). The baby dragon doesn't help you mine, the young and adult dragon's breath attack will break dirt and stone tiles, but they will only target enemies (and given how much food their attack takes, it'd be a waste to have them mine).
As for big dragons following you, this is because of their size. If you mine out a large shaft (or have mined out the whole island), they will follow you underground, but they can't fit into little tunnels. You can also teleport them by unequipping and re-equipping them if you have entered a large open space underground. They won't appear when you do this if the space is too small (and the adult dragons need a LOT of space).