  We study classical stochastic systems with discrete states, coupled to
switching external environments. For fast environmental processes we derive
reduced dynamics for the system itself, focusing on corrections to the
adiabatic limit of infinite time scale separation. In some cases, this leads to
master equations with negative transition `rates' or bursting events. We devise
a simulation algorithm in discrete time to unravel these master equations into
sample paths, and provide an interpretation of bursting events. Focusing on
stochastic population dynamics coupled to external environments, we discuss a
series of approximation schemes combining expansions in the inverse switching
rate of the environment, and a Kramers--Moyal expansion in the inverse size of
the population. This places the different approximations in relation to
existing work on piecewise-deterministic and piecewise-diffusive Markov
processes. We apply the model reduction methods to different examples including
systems in biology and a model of crack propagation.
