  We study the glass transition by exploring a broad class of kinetic rules
that can significantly modify the normal dynamics of super-cooled liquids,
while maintaining thermal equilibrium. Beyond the usual dynamics of liquids,
this class includes dynamics in which a fraction $(1-f_R)$ of the particles can
perform pairwise exchange or 'swap moves', while a fraction $f_P$ of the
particles can only move along restricted directions. We find that (i) the
location of the glass transition varies greatly but smoothly as $f_P$ and $f_R$
change and (ii) it is governed by a linear combination of $f_P$ and $f_R$.
(iii) Dynamical heterogeneities are not governed by the static structure of the
material. Instead, they are similar at the glass transition across the ($f_R$,
$f_P$) diagram. These observations are negative items for some existing
theories of the glass transition, particularly those reliant on growing
thermodynamic order or locally favored structure, and open new avenues to test
other approaches.
