  Consider continuous-time random walks on Cayley graphs where the rate
assigned to each edge depends only on the corresponding generator. We show that
the limiting speed is monotone increasing in the rates for infinite Cayley
graphs that arise from Coxeter systems, but not for all Cayley graphs. On
finite Cayley graphs, we show that the distance -- in various senses -- to
stationarity is monotone decreasing in the rates for Coxeter systems and for
abelian groups, but not for all Cayley graphs. We also find several examples of
surprising behaviour in the dependence of the distance to stationarity on the
rates. This includes a counterexample to a conjecture on entropy of Benjamini,
Lyons, and Schramm. We also show that the expected distance at any fixed time
for random walks on $\mathbb{Z}^+$ is monotone increasing in the rates for
arbitrary rate functions, which is not true on all of $\mathbb{Z}$. Various
intermediate results are also of interest.

Probability