The recent stuff I have seen is negative results:

(a) Can't assign Pearlian semantics to cyclic graphs.

(b) If you assign equilibrium semantics, you might as well use a dynamic causal Bayesian network, a cyclic graph does not buy you anything.

(c) A graph representing the Markov property of the equilibrium distribution of a Markov chain represented by a causal DBN is an interesting open question. (This graph wouldn't have a causal interpretation of course).