Labelled Markov Processes (LMP) seem to be a generalization of Probabilistic Automata (PA) studied by Segala to the case of the general state space. Namely, any LMP is given by a be a finite set of labels $A$, a measurable state space $(X,\mathscr B(X))$ and substochastic kernels $$ \forall a\in A,\;\tau_a:X\times \mathscr B(X)\to [0,1]. $$ In case $X$ is finite and $\mathscr B(X) = 2^X$, we obtain simply a PA.
The behavior of the PA is clear to me: it is given by trace distributions over the sequences of labels - e.g. in Section 3 here. I would expect the very same definition of the behavior for LMP, but I have never seen it defined formally anywhere. I would appreciate if you can hint upon a source describing the behaviour of a LMP.
Updated: after a short discussion, I have to clarify some details. I am familiar with measure-theoretical approach to probability and stochastic processes, and I think that behavioral semantics of PA extends directly to LMP modulo dealing with Borel or universally measurable transition kernels to characterize an adversary. However, I am not sure whether this is a possible way to define a behavior of LMP, or it is the intended one. For example, it may perhaps be described in the PhD thesis of J. Desharnais (which I do not have an access to).