  We study a general class of interacting particle systems called kinetically
constrained models (KCM) in two dimensions tightly linked to the monotone
cellular automata called bootstrap percolation. There are three classes of such
models, the most studied being the critical one.
  Together with the companion paper by Mar\^ech\'e and the author, our work
determines the logarithm of the infection time up to a constant factor for all
critical KCM, which were previously known only up to logarithmic corrections.
We establish that on this level of precision critical KCM have to be classified
into seven categories refining the two in bootstrap percolation and the two in
previous rougher results. In the present work we establish the upper bounds for
the novel five categories and thus complete the universality program for
equilibrium critical KCM. Our main innovations are the identification of the
dominant relaxation mechanisms and a more sophisticated and robust version of
techniques recently developed for the study of the Fredrickson-Andersen 2-spin
facilitated model.
