  A partial order on the set of metric measure spaces is defined; it
generalizes the Lipschitz order of Gromov. We show that our partial order is
closed when metric measure spaces are equipped with the Gromov-weak topology
and give a new characterization for the Lipschitz order. We will then consider
some probabilistic applications. The main importance is given to the study of
Fleming-Viot processes with different resampling rates. Besides that
application we also consider tree-valued branching processes and two semigroups
on metric measure spaces.
