This kind of process is sometimes called random exchange process. A starting point for a literature research might be the following article by Helland and Nilsen:
For a recent work with a characterization of nullrecurrence see
Addendum: Kellerer has studied order-preserving Markov chains in the following paper.
The invariant distribution and its existence is independent of the starting distribution, at least under slight irreducibility conditions. In his comment after Theorem 10.1 Kellerer gives a characterization for finite first moment of an invariant distribution for exchange processes. For studying return times and their expectation values, his work in Chapter 11 might be helpful. For example, when $X_1$ is bounded, I believe that Theorem 11.2 and Proposition 11.3 directly imply finiteness of the expectation of all nondegenerate return times (possibly this is a rather trivial result). When $X_1$ is not bounded, things are more difficult of course.