  The possibilities to accelerate vortex electrons with orbital angular
momentum (OAM) to relativistic energies and to produce vortex ions, protons,
and other charged particles crucially depend on whether the OAM is conserved
during the acceleration and on how phase space of the wave packet evolves. We
show that both the OAM and a mean emittance of the packet, the latter obeying
the Schr\"odinger uncertainty relation, are conserved in axially symmetric
fields of electric and magnetic lenses, typical for accelerators and electron
microscopes, as well as in Penning traps, while a linear approximation of
weakly inhomogeneous fields works much better for single packets than for
classical beams. We analyze quantum dynamics of the packet's rms radius
$\langle\rho^2\rangle$, relate this dynamics to a generalized form of the van
Cittert-Zernike theorem, applicable at arbitrary distances from a source and
for non-Gaussian packets, and adapt the Courant-Snyder formalism to describe
the evolution of the wave packet's phase space. The vortex particles can
therefore be accelerated, focused, steered, trapped, and even stored in
azimuthally symmetric fields and traps, somewhat analogously to the classical
angular-momentum-dominated beams. Moreover, we give a quantum version of the
Busch theorem, which states how one can produce vortex electrons with a
magnetized cathode during either field- or photoemission, as well as vortex
ions and protons by using a magnetized stripping foil employed to change a
charge state of ions. Spatial coherence of the packets plays a crucial role in
these applications and we provide the necessary estimates for particles of
different masses.
