Dredging through long-ago memories of my time in a laser lab...
One critical reason for pulsed beams for cutting or weaponry lasers is to allow literal physical detritus to exit the beampath - there's a nontrivial outpuffing of small particles, splinters and fragments, and vapour of the target material at the beam impact point as the surface reaches a boil-off temperature - you get micro-spalling and micro-explosions driving these fragments outwards - a pulsed beam means each pulse passes through largely unobstructed space to impact the target surface.
Another nontrivial reason is to do with the sheer amount of energy being passed through the beam - depending upon wavelength being emitted and the atmospheric content (water vapour, particulates etc) you can get instant ionisation of the air in the beampath - and this can then become an energised and ionised plasma channel pretty quickly. Whilst this looks impressive as hell, (and in fact can become a whole new category of weapon with sufficient modification [see beamed lighting weapons]) it plays absolute hell with your beam actually reaching its target... so pulsing helps obviate that issue by limiting the quanta of energy absorbed per unit time per unit atmospheric volume.
I once saw a very large neodymium YAG laser (2" beam apeture & 8 Watts continuous beam power) that had to be fired at the target (was being used to instantly slag very thick glass and melt in a fluorescent dye) through tubes filled with argon - they discovered during an early test firing at a test target in open atmosphere that the beam generated enough ozone, and then ionised it, and superheated it, all instantly, that they observed what appeared to be a "beam of fire" along the beampath - scared heck out of them, safety shutdown, long time to recondition the laser for next use - hence the argon isolation in all future attempts.
Pulsed lasers also avoid any heat buildup issues in the actual laser equipment, for the record.
Edit post this comment: "...So the same idea can be applied to particle accelerators in terms of heat management? ..."
I don't think you can draw that inference with any confidence, no - I think it depends entirely on how your accelerator is designed. Consider the conceptual and technical differences between a full cyclotron and a cavity magnetron - both were used (briefly) in early microwave ovens as the particle accelerator (kylstrons were used for larger microwave arrays) or heck include in this list the more modern equivalent of the cyclotron - the synchrotron - they all fall under the gerenal category of "particle accelerator", and each is technically vastly different.
Cyclotrons and synchrotrons spin the particles round (synchros are a true circle, the cyclotron is loosely speaking a spiral) Klystrons are linear... magnetrons use cavity resonance to generate microwaves (like blowing over a bottle lip to make that resonant whistle) and so they all loosely do similar things, each is vastly different in working principle, underlying technology, power requirements, continuity of power use, and conversion efficiency.
So for example, anything which is synchro or cyclo based will not get any benefit in heat management from a pulsed beam, as the particle accelerator would be running continuously, and only intermittently allowing egress of particles... whereas a linear accelerator, or some super high tech solid state klystron equivalent would benefit from an oscillating output.
Caveat to all this: I am NOT an electrical engineer, thermionics engineer or resonance engineer, and my exposure to Klystrons, magnetrons, cyclotrons, synchrotrons et al are all based on growing up with a father who was an electrical engineer and thermionics engineer. So I grew up with vacuum valves around the house, and frequent discussions of the history of thermionics development and the use of such devices in experimental tropospheric scatter radio stations in the pre-satellite era attempt to get beyond horizon communications links developed.