We study the motion of a classical relativistic charged particle in the electromagnetic field of a Bessel beam exhibiting vector optical vortices, and show how its dynamical properties, such as linear and angular momentum, are transmitted to the particle. The effects of different polarizations are taken into account using transverse electric and magnetic modes, and their superpositions. The constants of motion are identified for the most general case. We report typical numerical results for axial and radial motion for various configurations, with an estimate of expected axial accelerations when transverse magnetic Bessel modes are used. The Lorentz transformation properties of the field are used throughout in order to simplify the calculations.