  For a matroid with an ordered (or "labelled") basis, a basis exchange step
removes one element with label $l$ and replaces it by a new element that
results in a new basis, and with the new element assigned label $l$. We prove
that one labelled basis can be reconfigured to another if and only if for every
label, the initial and final elements with that label lie in the same connected
component of the matroid. Furthermore, we prove that when the reconfiguration
is possible, the number of basis exchange steps required is $O(r^{1.5})$ for a
rank $r$ matroid. For a graphic matroid we improve the bound to $O(r \log r)$.
