Density-functional calculations of lattice dynamics and high-resolution synchrotron powder diffraction uncover antiferroelectric distortion in the kagome francisite Cu$_3$Bi(SeO$_3$)$_2$O$_2$Cl below 115K. Its Br-containing analogue is stable in the room-temperature crystal structure down to at least 10K, although the Br compound is on the verge of a similar antiferroelectric instability and reveals local displacements of Cu and Br atoms. The I-containing compound is stable in its room-temperature structure according to density-functional calculations. We show that the distortion involves cooperative displacements of Cu and Cl atoms, and originates from the optimization of interatomic distances for weakly bonded halogen atoms. The distortion introduces a tangible deformation of the kagome spin lattice and may be responsible for the reduced net magnetization of the Cl compound compared to the Br one. The polar structure of Cu$_3$Bi(SeO$_3$)$_2$O$_2$Cl is only slightly higher in energy than the non-polar antiferroelectric structure, but no convincing evidence of its formation could be obtained.
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