As I seasoned brewer, but novice distiller, I have seen dozens of systems in place for grain handling, I would suggest the following....
Pneumatic - Less "damage" to the material being transported (more important with lautering barley mashes). Noisy and typically costly. Can be dusty if used with fine ground material, but not always. Usually works with few issues and minimal maintenance. Usually a better solution for larger operations.
Screw Auger - A bit more damage to the material, but economical and relatively easy to install. Can have issues moving all of the finely ground materials depending on the elevations, etc. Not bad on maintenance, but can bind/buckle if not installed properly. Usually a good compromise for most operations.
Cable/Chain Disc - Less damage to the material, but a bit bulkier/more complex to install. It moves most all material cleanly due to a cellular/peristaltic type design. Can be finicky on alignments and maintenance, sometimes. Works great for in-line additions of materials (blending).
A relatively cheap option may be a "gran suction machine" from China....Its basically a small screw auger with a fixed or flexible transport tube. Another DIY solution could be a leaf blower with PVC fittings - "corn blower". Standard disclaimer, don't forget that grain dust is extremely flammable/explosive, and is just as dangerous as high proof spirits.
Some may feel differently on this one - For mills, the previous is true. Hammer mills are better for corn, and probably rye and wheat kernels as well, compared to barley - however - this is primarily due to their ability to pulverize and expose the starch for better hydration/enzymatic action. IF you are planning to use exogenous enzymes to help starch conversion, I would suggest a hammer mill offers only the smallest advantage. The design of the hammer mill usually makes it a bit more reliable over long term (commercial-type) usage, where the knurled rollers on a roller mill will need re-worked over a period of time. Either machine is subject to bearings/motors/no grease/etc. Roller mills used to be a more economical option, but the market has leveled some.