  The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics
(or variational) model for a nonconserved order parameter field. After revising
main literature results for the occuring different types of moving fronts, we
employ path continuation to determine their bifurcation diagram in dependence
of the external field strength or chemical potential. We then employ the same
methodology to systematically analyse fronts for more involved AC-type models.
In particular, we consider a cubic-quintic variational AC model and two
different nonvariational generalisations. We determine and compare the
bifurcation diagrams of front solutions in the four considered models.
