High-pressure structural phase transitions in NaNiF3 and NaCoF3 were investigated by conducting in situ synchrotron powder X-ray diffraction experiments using a diamond anvil cell. The perovskite phases (GdFeO3 type) started to transform into postperovskite phases (CaIrO3 type) at about 11-14 GPa, even at room temperature. The transition pressure is much lower than those of oxide perovskites. The anisotropic compression behavior led to heavily tilted octahedra that triggered the transition. Unlike oxide postperovskites, fluoropostperovskites remained after decompression to 1 atm. The postperovskite phase in NaCoF3 broke down into a mixture of unknown phases after laser heating above 26 GPa, and the phases changed into amorphous ones when the pressure was released. High-pressure and high-temperature experiments using a multianvil apparatus were also conducted to elucidate the phase relations in NaCoF3. Elemental analysis of the recovered amorphous samples indicated that the NaCoF3 postperovskite disproportionated into two phases. This kind of disproportionation was not evident in NaNiF3 even after laser heating at 54 GPa. In contrast to the single postpostperovskite phase reported in NaMgF3, such a postpostperovskite phase was not found in the present compounds.