Let μ be a point on a countable discrete space [InlineMediaObject not available: see fulltext.]. Under the assumption that μ is quasi-invariant with respect to any finitary permutation of [InlineMediaObject not available: see fulltext.], we describe a general scheme for constructing an equilibrium Kawasaki dynamics for which μ is a symmetrizing (and hence invariant) measure. We also exhibit a two-parameter family of point process μ possessing the needed quasi-invariance property. Each process of this family is determinantal, and its correlation kernel is the kernel of a projection in ℓ2 ([InlineMediaObject not available: see fulltext.]). Bibliography: 17 titles.