The latter have abacuses which are used to programme co-ordinates for the desired destination. The branches and root systems are linked all over the magical world to these abacuses. The tree itself doesn't move (not from looking from the outside) but the inside of the tree "shakes" representing movement as powerful magic pulsates through the tree to make the thing work. It is not a smooth ride to put it mildly though Eileen is expert at programming the trees so as to get the best ride possible.
Most magical beings can fly under their own steam though the sprites have to wear special boots which they tap and the distance they can go is limited. This is partly done by superior magical species to limit the havoc sprites are known to cause given their greatest loves are drinking and fighting.
The skies in the Kingdom get pretty crowded at rush hour times and collisions are not unknown. Despite government information films put out by FNN regularly, nobody has shown any signs of wanting to change the way they fly. This does not surprise Eileen though it does make the Queen and her Council despair.