Markov chains and mappings of distributions on compact spaces

Consider a compact metric space S and a pair $(j,k)$ with k ≥ 2 and 1 ≤ j ≤ k. For any probability distribution θ ∈ P(S), define a Markov chain on S by: from state s, take k i.i.d. (θ) samples, and jump to the j'th closest. Such a chain converges in distribution to a unique stationary distribution, say πj,k(θ). So this defines a mapping πj,k: P(S) → P(S). What happens when we iterate this mapping? In particular, what are the fixed points of this mapping? We present a few rigorous results, to complement our extensive simulation study elsewhere.