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 and wairakite contains Ca. Under compression, all three minerals display structural phase transitions to lower-symmetry forms. In cubic analcime the phase transition occurs at a relatively low pressure (similar to 1 GPa). Our recent theoretical Study using geometric simulation indicates that this phase transition is controlled by the flexibility window of the ANA framework. The flexibility window is defined as the range of "framework densities" over which the tetrahedral units of the framework can in principle be made geometrically ideal. In leucite and wairakite the ambient-pressure structure is more distorted than in analcime, due to the different cation content, and their subsequent phase transitions occur at higher pressures (similar to 2-3 GPa). We discuss the experimental data for these minerals and its relationship to the theoretical flexibility of the ANA framework and to the influence of cation content.