Will joint compound / drywall mud clog my drain?
I think it would depend on the amount of compound you're putting down the drain, what size drain, how much water is used to flush the tools, and probably a few other factors I haven't thought of. Though typically it's not a problem, as long as you're not pouring a whole tray of mud down the drain.

Scoop out any left over mud from your tray/hawk, and throw it away. Then rinse all the tools thoroughly. 

NOTE: If you're using premixed compound, never put unused mud back in the bucket.  Once the mud is out of the bucket, either use it, or toss it.