We study basic properties of flow equivalence on one-dimensional
compact metric spaces with a particular emphasis on isotopy in the
group of (self-) flow equivalences on such a space. In particular, we
show that such an orbit-preserving map is not always an isotopy,
but that this always is the case for suspension flows of irreducible
shifts of finite type. We also provide a version of the fundamental
discretization result of Parry and Sullivan which does not require that
the flow maps are either injective or surjective. Our work is motivated
by applications in the classification theory of sofic shift spaces, but
has been formulated to supply a solid and accessible foundation for
other purposes.