In this paper we derive intertwining relations for a broad class of conservative particle systems both in discrete and continuous setting. Using the language of point process theory, we are able to derive a new framework in which duality and intertwining can be formulated for particle systems evolving in general spaces. These new intertwining relations are formulated with respect to factorial and orthogonal polynomials.

Our novel approach unites all the previously found self-dualities in the context of discrete consistent particle systems and provides new duality results for several interacting systems in the continuum, such as interacting Brownian motions. We also introduce a process that we call generalized inclusion process, consisting of interacting random walks in the continuum, for which our method applies and yields generalized Meixner polynomials as orthogonal self-intertwiners.