Cube‐like double four‐ring (d4r) cages are among the most frequent building units of zeolites and zeotypes. In materials synthesised in fluoride‐containing media, the fluoride anions are preferentially incorporated in these cages. In order to study the impact of framework composition and organic structure‐directing agents (OSDAs) on the possible occurrence of local distortions of fluoride‐containing d4r cages, density functional theory (DFT) calculations and DFT‐based molecular dynamics simulations were performed for AST‐type zeotypes, considering four different compositions (SiO2, GeO2, AlPO4, GaPO4) and two different OSDA cations (tetramethylammonium [TMA] and quinuclidinium [QNU]). All systems except SiO2‐AST show significant deformations, with a pyritohedron‐like distortion of the d4r cages occurring in GeO2‐ and GaPO4‐AST, and a displacement of the fluoride anions towards one of the corners of the cage in AlPO4‐ and GaPO4‐AST. While the distortions occur at random in TMA‐containing zeotypes, they exhibit a preferential orientation in systems that incorporate QNU cations. In addition to providing detailed understanding of the local structure of a complex host‐guest system on the picosecond timescale, this work indicates the possibility to stabilise ordered distortions through a judicious choice of the OSDA, which might enable a tuning of the material's properties.