  From the Poisson-Dirichlet diffusions to the $Z$-measure diffusions, they all
have explicit transition densities. In this paper, we will show that the
transition densities of the $Z$-measure diffusions can also be expressed as a
mixture of a sequence of probability measures on the Thoma simplex. The
coefficients are still the transition probabilities of the Kingman coalescent
stopped at state $1$. This fact will be uncovered by a dual process method in a
special case where the $Z$-measure diffusions is established through up-down
chain in the Young graph.
