Question: Is it possible to "pump" a powder the same way liquids can be pumped?

If so, what are the challenges? If not, what are some alternatives?

Answer: 
There is a key difference between liquids and powders:

* Liquids have internal cohesion and can be sucked, but with a powder, you need a carrying medium.
* The internal friction in a powder will usually be far higher than in a liquid.

In practice, this means that to pump a powder, you inject air or another gas as fluidizing agent. The pumped or blown air carries the powder, like a dust-storm within the pipe.
Yes you can pump powders using a diaphragm pump. The above comments all focus on alternate ways to transport powder they are not talking about pumping. Check out powder pumping with a diaphragm pump on youtube.