As electrons are accelerated, emitted photons take with them certain part of electron's momentum. So, radiation can cause both dumping or undumping of electron oscilations in any particular direction. It is possible, by particular accelerator lattice design, to acheive damping in all three directions.
Because of the quantum effects - electrons radiate statistically - dumping (ie. beam becoming smaller and smaller) process will not continue indefinitely. Energy spread of the beam have small but finite size determined by the finite size of Plank's constant.
Recently, Huang , Chen and Ruth showed that in uniform focusing (linear) channel radiation damping is much slower but final emittance may be significantly lower as there is no continual photon emission (no circular path).