Equilibrium Kawasaki dynamics and determinantal point processes

Let $\mu$ be a point process on a countable discrete space $\mathfrak X$. Under the assumption that $\mu$ is quasi-invariant with respect to any finitary permutation of $\mathfrak X$, we describe a general scheme for constructing an equilibrium Kawasaki dynamics for which $\mu$ is a symmetrizing (and hence invariant) measure. We also exhibit a two-parameter family of point processes $\mu$ possessing the needed quasi-invariance property. Each process of this family is determinantal, and its correlation kernel is the kernel of a projection operator in $\ell^2(\mathfrak X)$.