  We give conditions under which near-critical stochastic processes on the
half-line have infinitely many or finitely many cutpoints, generalizing
existing results on nearest-neighbour random walks to adapted processes with
bounded increments satisfying appropriate conditional increment moments
conditions. We apply one of these results to deduce that a class of transient
zero-drift Markov chains in $\mathbb{R}^d$, $d \geq 2$, possess infinitely many
separating annuli, generalizing previous results on spatially homogeneous
random walks.
