  We study the full class of kinetically constrained models in arbitrary
dimension and out of equilibrium, in the regime where the density $q$ of
facilitating sites in the equilibrium measure (but not necessarily in the
initial measure) is close to $1$. For these models, we establish exponential
convergence to equilibrium in infinite volume and linear time precutoff in
finite volume with appropriate boundary condition. Our results are the first
out-of-equilibrium results that hold for any model in the so-called critical
class, which is covered in its entirety by our treatment, including e.g. the
Fredrickson-Andersen 2-spin facilitated model. In addition, they generalise,
unify and sometimes simplify several previous works in the field. As byproduct,
we recover and generalise exponential tails for the connected component of the
origin in the upper invariant trajectory of perturbed cellular automata and in
the set of eventually infected sites in subcritical bootstrap percolation
models. Our approach goes through the study of cooperative contact processes,
last passage percolation, Toom contours, as well as a very convenient coupling
between contact processes and kinetically constrained models.
