  The effect of a finite width of a particle bunch on radiation of twisted
photons is studied. The general formulas connecting the radiation probability
distribution of twisted photons produced by bunches of identical particles with
the radiation probability distribution of twisted photons generated by one
particle are obtained for axially symmetric bunches. The bunch is called
axially symmetric if it is axially symmetric with respect to the detector axis
at some instant of time and all the particles in the bunch move along parallel
trajectories. The general sum rules for the probability of radiation of twisted
photons by axially symmetric bunches are established. In particular, we prove
that the projection of the average total angular momentum of radiated twisted
photons per particle in the bunch does not depend on the radial profile of the
bunch. The uniform, Gaussian, and exponential radial bunch profiles are
considered in detail. The radiation of axially symmetric bunches in ordinary
and crystalline undulators is investigated. The selection rules for radiation
of twisted photons by one particle in undulators are violated when the finite
width of the particle bunch is taken into account. We find the condition when
this violation is marginal. The form of the radiation probability distribution
of twisted photons becomes universal for wide incoherent axially symmetric
particle bunches. We completely describe these universal distributions.
