There are two classes of artifacts which grow without limit, in the sense that it can be demonstrated that the number of live cells in the field will always continue to increase. The first to be discovered consisted of the glider guns, for which a continuous stream of gliders emerged from a stationary region; the readily apparent periodicity of the process constituted the element of proof required.
An alternative would be for a moving source to leave behind a trail of stationary objects. Gliders are too small and fragile to serve the purpose, but other artifacts are suitable. Space ships, as they are called, are seen, albeit infrequently, in the residue evolving from random initial configurations; they differ from gliders by moving along the coordinate axes instead of diagonally. Nevertheless, they involve a reflective stage, managing to advance by two cells every four generations.
Figure: A space ship that moves two cells to the right in four
generations, shown in two of its four phases.
In common with complex oscillators, there are some sparks associated with the movement of a space ship which are capable of interacting with other objects without impeding the motion of the space ship. Quite a large variety of followers can be constructed, many of which leave an orderly assemblage of residues in their wake; this is the combination which is called a puffer train.
There are even assemblages which leave no residue --- smokeless puffer trains --- which are really space ships themselves; the majority leave an untidy mess which may even take a life of its own, advance rapidly, and destroy the engine.