Spreading maps (polymorphisms), symmetries of Poisson processes, and matching summation

The matrix of a permutation is a partial case of Markov transition matrices. In the same way, a measure preserving bijection of a space $(A,\alpha)$ with finite measure is a partial case of Markov transition operators. A Markov transition operator also can be considered as a map (polymorphism) $(A,\alpha)\to (A,\alpha)$, which spreads points of $(A,\alpha)$ into measures on $(A,\alpha)$.
Denote by $\mathbb R^*$ the multiplicative group of positive real numbers and by $\mathscr M$ the semigroup of measures on $\mathbb R^*$. In this paper, we discuss $\mathbb R^*$-polymorphisms and $\curlyvee$-polymorphisms, who are analogues of the Markov transition operators (or polymorphisms) for the groups of bijections $(A,\alpha)\to (A,\alpha)$ leaving the measure $\alpha$ quasiinvariant; two types of the polymorphisms correspond to the cases, when $A$ has finite and infinite measure respectively. For the case, when the space $A$ itself is finite, the $\mathbb R^*$-polymorphisms are some $\mathscr M$-valued matrices.
We construct a functor from $\curlyvee$-polymorphisms to $\mathbb R^*$-polymorphisms, it is described in terms of summations of $\mathscr M$-convolution products over matchings of Poisson configurations.