Distances between realizations of order types

Any $n$-tuple of points in the plane can be moved to any other $n$-tuple by a continuous motion with at most $\binom{n}{3}$ intermediate changes of the order type. Even for tuples with the same order type, the cubic bound is sharp: there exist pairs of $n$-tuples of the same order type requiring $c\binom{n}{3}$ intermediate changes.