The far-infrared magnetotransmission of thin films of semiconducting and semimetallic ${\mathrm{Bi}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sb}}_{\mathit{x}}$ alloys grown by molecular-beam epitaxy has been measured at fixed photon energies between 2.5 and 21.4 meV in magnetic fields up to 6 T, at T=1.8 K. The samples, grown on ${\mathrm{BaF}}_{2}$ substrates with composition 0\ensuremath{\le}x\ensuremath{\le}22.5%, were monocrystalline, with the trigonal axis perpendicular to the surface plane. The measurements were carried out in Faraday and Voigt geometries, with the magnetic field oriented parallel to binary, bisectrix, and trigonal axes of the films. Cyclotron-resonance lines of both electrons and holes were observed. From them, we establish the composition dependence of the effective-mass tensor, of the direct L-point band gap, and of the energy overlap in the semimetallic samples. We conclude that all band-structure parameters are the same in the films as in bulk ${\mathrm{Bi}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sb}}_{\mathit{x}}$ alloys, except for the energy overlap, which is increased by 16 meV independently of composition, possibly because of the strain induced by the substrate.