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: 
I believe it's termed "injecting" not "pumping" but yes, it's certainly possible.

Coal fired electric power plants grind the coal to a very fine dust and then blow the dust into the central burner.

[This particular article](http://www.netl.doe.gov/File%20Library/Research/Coal/major%20demonstrations/cctdp/Round3/bethstl.pdf) is a summary of some DOE research for a Blast Furnace Granulated Coal Injection System Demonstration Project.

And [this article](http://www.nfpa.org/~/media/Files/Research/Research%20Foundation/foundation%20proceedings/david_snell.pdf) discusses some of the hazards from a National Fire Protection Association perspective. Coal dust having the particular dangerous property of being explosive and all that...

As far as specifics regarding danger, I suspect it will vary depending upon the material(s) involved in the powder.
You may also look into pneumatic conveying. It involves [fluidization of the particles](http://en.wikipedia.org/wiki/Fluidization). 

This simply uses pressure differential to move any bulk material. It's used a lot in industrial applications. You can read a [summary of those systems, here](http://www.nol-tec.com/answers-to-eight-common-questions.html).