We present theoretical and numerical results on the Compton scattering of a twisted Bessel photon by a plane-wave electron with an arbitrary initial velocity. Scattering of a photon prepared in a superposition of Bessel states is also explored. Specifically, we investigate how the electron motion affects the dependence of the angular distribution and polarization of the scattered photons on the twisted photon opening angle, projection of angular momentum, and on the superposition of angular-momentum projection. The motion of the electron is found to strongly influence the dependence of these quantities on the parameters of the twisted photon. Moreover, if the speed of a properly directed electron exceeds a critical value, various distributions, such as the degree of circular polarization of the scattered photons, undergo a nearly global reversal of sign that is absent from the scattering of a plane-wave photon under identical conditions. This behavior is found to correspond to the inversion of the twisted photon momentum cone in the rest frame of the electron.