In this article we review the important role of non-equilibrium dynamics in reactions induced by ions and hadron beams to understand the fragmentation processes inside hadronic media. We discuss the single-particle dynamics in specific sources such as spectators in heavy-ion collisions and residual nuclear targets in hadron-induced reactions. Particular attention is given to the dynamics of hyperons. We further discuss the question regarding the onset of local instabilities, which are relevant for the appearance of fragmentation phenomena in nuclear reactions. We apply the theoretical formalism, that is, semi-classical transport embedded with statistical methods of nuclear fragmentation, to reactions induced by light ions and hadron beams. We discuss the results of nuclear fragmentation and, in particular, examine the formation of hypernuclei. Such studies are important for obtaining a deeper understanding of the equation of state in fragmenting matter and are relevant for forthcoming experiments, such as PANDA at FAIR and J-PARC in Japan.