  Mapping the behaviour of dark energy is a pressing task for observational
cosmology. Phenomenological classification divides dynamical dark energy models
into freezing and thawing, depending on whether the dark energy equation of
state is approaching or moving away from $w=p/\rho=-1$. Moreover, in realistic
dynamical dark energy models the dynamical degree of freedom is expected to
couple to the electromagnetic sector, leading to variations of the
fine-structure constant $\alpha$. We discuss the feasibility of distinguishing
between the freezing and thawing classes of models with current and forthcoming
observational facilities and using a parametrisation of the dark energy
equation of state, which can have either behaviour, introduced by Mukhanov as
fiducial paradigm. We illustrate how freezing and thawing models lead to
different redshift dependencies of $\alpha$, and use a combination of current
astrophysical observations and local experiments to constrain this class of
models, improving the constraints on the key coupling parameter by more than a
factor of two, despite considering a more extended parameter space than the one
used in previous studies. We also briefly discuss the improvements expected
from future facilities and comment on the practical limitations of this class
of parametrisations. In particular, we show that sufficiently sensitive data
can distinguish between freezing and thawing models, at least if one assumes
that the relevant parameter space does not include phantom dark energy models.
