We consider exchangeable Markov multi-state survival processes -- temporal processes taking values over a state-space$\mathcal{S}$ with at least one absorbing failure state $\flat \in \mathcal{S}$ that satisfy natural invariance properties of exchangeability and consistency under subsampling. The set of processes contains many well-known examples from health and epidemiology -- survival, illness-death, competing risk, and comorbidity processes; an extension leads to recurrent event processes. We characterize exchangeable Markov multi-state survival processes in both discrete and continuous time. Statistical considerations impose natural constraints on the space of models appropriate for applied work. In particular, we describe constraints arising from the notion of composable systems. We end with an application of the developed models to irregularly sampled and potentially censored multi-state survival data, developing a Markov chain Monte Carlo algorithm for posterior computation.