I explore the dynamics of small dust particles in transport-dominated circumstellar debris discs in binary star systems. In these tenuous discs the effects of mutual collisions are negligible and their morphology is determined by Poynting–Robertson drag and, possibly, by the strong perturbations induced by the interaction with the interstellar medium (ISM) flux of neutral atoms. The force due to the ISM flux can significantly affect the dynamical behaviour of the dust grains, causing a fast inward drift and a large periodic oscillation of both eccentricity and inclination. If the disc is around a star in a binary system, the gravity of the companion star competes with the ISM force and the dynamics is complex. The balance between the two forces depends strongly on the binary semimajor axis aB and eccentricity eB. In a scenario with an ISM flux similar to that observed in the Solar system neighbourhood, the binary secular perturbations, assuming a mass ratio of 0.5, dominate over the ISM force when aB < 600 au and eB= 0.2. This occurs when the dust disc is generated by a parent body ring encompassed between 50 and 60 aufrom the primary star. For a larger binary eccentricity eB= 0.6, the limit moves to aB < 700 au.
Within these values of aB, the time-scale of the binary secular perturbations is much shorter than the period of the ISM-induced orbital variations, and the disc shape and density distribution are dominated by the companion gravity. It appears slightly eccentric and, if the binary is coplanar with the disc, only a limited warping due to the ISM perturbations is observed. In this scenario, the strong ISM perturbations, which may significantly affect debris discs around single stars embedded in strong ISM winds, are almost completely silenced.
For larger semimajor axes, the scenario is reversed with the ISM perturbations ruling the dynamics of the dust. The disc develops a large clump oriented at 90° with respect to the direction of the ISM flux and it appears highly warped in space. Since the period of the two perturbations can be analytically estimated, it is possible to deduce by easy computations when a given system is governed either by the binary gravity or by the ISM perturbations.