The theory of probabilistic dynamics (TPD) offers a framework capable of modeling the interaction between the physical evolution of a system in transient conditions and the succession of branchings defining a sequence of events. Nonetheless, the Chapman-Kolmogorov equation, besides being inherently Markovian, assumes instantaneous changes in the system dynamics when a setpoint is crossed. In actuality, a transition between two dynamic evolution regimes of the system is a two-phase process. First, conditions corresponding to the triggering of a transition have to be met; this phase will be referred to as the activation of a "stimulus." Then, a time delay must elapse before the actual occurrence of the event causing the transition to take place. When this delay cannot be neglected and is a random quantity, the general TPD can no longer be used as such. Moreover, these delays are likely to influence the ordering of events in an accident sequence with competing situations, and the process of delineating sequences in the probabilistic safety analysis of a plant might therefore be affected in turn. This paper aims at presenting several extensions of the classical TPD, in which additional modeling capabilities are progressively introduced. A companion paper sketches a discretized approach of these problems.