We report on the observation of a terahertz radiation induced photon drag effect in epitaxially grown $n$- and $p$-type (Bi$_{1-x}$Sb$_{x}$)$_{2}$Te$_{3}$ three dimensional topological insulators with different antimony concentrations $x$ varying from 0 to 1. We demonstrate that the excitation with polarized terahertz radiation results in a $dc$ electric photocurrent. While at normal incidence a current arises due to the photogalvanic effect in the surface states, at oblique incidence it is outweighed by the trigonal photon drag effect. The developed microscopic model and theory show that the photon drag photocurrent is due to the dynamical momentum alignment by time and space dependent radiation electric field and implies the radiation induced asymmetric scattering in the electron momentum space.