The analcime-like feldspathoids are a group of microporous minerals with the ANA framework topology. Analcime proper has predominantly Na as its channel cation content, while leucite contains predominantly K, pollucite contains Cs and wairakite contains Ca. Under compression, all these minerals display structural displacive phase transitions to low-symmetry forms at relatively low pressures (0.5-3.2 GPa). We show, using geometric simulation, that these phase transitions are controlled by the "flexibility window" of the ANA framework, defined as the range of framework densities over which the tetrahedral units of the framework can in principle be made geometrically ideal. We discuss experimental compression data for these minerals and its relationship to the theoretical flexibility of the ANA framework, the influence of cation content, and framework ordering.