... the laser itself is fixed in position and mirrors are used to redirect the laser to the material.
Yes. This system is called "flying optics". The delicate laser is fixed in position and only the mirrors and lens move about. This simplifies the mechanical setup and reduces the inertia and required motor sizes.
Wouldn't the beam be wider at the edges of the machine's travel?
Figure 1. Divergence specification for Syntrad's ti60.
The beam divergence can be quite small. The Synrad unit above has < 7 mrad which would give a divergence of up to 7 mm/m of beam length. The optical path (mirrors, etc.) at maximum range would have to have large enough mirrors and the final collimation lens diameter would have to cope with the enlarged beam. The other course of action would be to correct for the divergence at source.
Figure 2. See Science ABC for example.