If I spray particles into a chaotic system, I would expect some tiny fraction of them to sling-shot around various attractive bodies, and some tiny fraction of those would be ejected to hit some other galaxy - like us.

It could be gravitational. Anything N>2 is unpredictable, with particles possibly exporting the total energy of the system, leaving everything else stationary (zero angular momentum, gently contracting). The same for electromagnetism.

Of course, there will be an occasional ultra-high-energy cosmic ray particle.

The only limit is the universal red-shift, which imposes a spectral shift on how far away this the particle could originate - but even then, it is just statistics, and it could be a 'lucky' high-energy escape from further away.