Behaviour of Labelled Markov Processes
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I think it depends on what you mean by behaviour. Probabilistic automata follow in the tradition of finite automata, so their behaviour is defined in terms of their language or traces. Labelled Markov Processes follow in the tradition of process algebra, where it is known that processes can be compared using a variety of preorders typically starting with bisimulation. You do have notions of bisimulation defined for LMPs and in that paper. Bisimulation is stronger than trace equivalence, as you may know, so the behaviour you are interested in is the tree unfolding rooted at a given state. Definition 3.1 of the paper you have linked gives approximations of this tree.