Previously, it has been established that axion dark matter (DM) is clustered to form clumps (axion miniclusters) with masses M ≈ 10–12M⊙. The passages of such clumps through the Earth are very rare events occurring once in 105 years. It has also been shown that the Earth’s passage through DM streams, which are the remnants of clumps destroyed by tidal gravitational forces from Galactic stars, is a much more probable event occurring once in several years. In this paper, we have performed detailed calculations of the destruction of miniclusters by taking into account their distribution in orbits in the Galactic halo. We have investigated two DM halo models, the Navarro–Frenk–White and isothermal density profiles. Apart from the Galactic disk stars, we have also taken into account the halo and bulge stars. We show that about 2–5% of the axion miniclusters are destroyed when passing near stars and transform into axion streams, while the clump destruction efficiency depends on the DM halo model. The expected detection rate of streams with an overdensity exceeding an order of magnitude is 1–2 in 20 years. The possibility of detecting streams by their tidal gravitational effect on gravitational-wave interferometers is also considered.