  The mixing behaviour of random walks on lattice points of polytopes using
Markov bases is examined. It is shown that under a dilation of the underlying
polytope, these random walks do not mix rapidly when a fixed Markov basis is
used. We also show that this phenomenon does not disappear after adding more
moves to the Markov basis. Avoiding rejections by sampling applicable moves
does also not lead to an asymptotic improvement. As a way out, a method of how
to adapt Markov bases in order to achieve the fastest mixing behaviour is
introduced.
