  We expand the dynamical systems investigation of cosmological scalar fields
characterised by kinetic corrections presented in [N. Tamanini, Phys. Rev. D 89
(2014) 083521]. In particular we do not restrict the analysis to exponential
potentials only, but we consider arbitrary scalar field potentials and derive
general results regarding the corresponding cosmological dynamics. Two specific
potentials are then used as examples to show how these models can be employed
not only to describe dark energy, but also to achieve dynamical crossing of the
phantom barrier at late times. Stability and viability issues at the classical
level are also discussed.
