We study the glass transition by exploring a broad class of kinetic rules that can significantly modify the normal dynamics of supercooled 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 move only 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 (DHs) are not governed by the static structure of the material; their magnitude correlates instead with the relaxation time. (iv) We show that a recent theory for temporal growth of DHs based on thermal avalanches holds quantitatively throughout 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, as we illustrate.