Wes, it depends.  What is most economic gets processed first, but sometimes the richest ores are left to keep the miner alive during hard times.  That means that sometimes the riches ores are stockpiled.  So the answer to your question is complicated by the size of the company/deposit, and the objectives of the BOD, as much as grades and economic cycles.  Then, too, some metals like selenium can at times have a negative worth such that stockpiling has to be done, but it is costly to keep it contained as it is an environmental poison.  I know you are aware of this given where you live.

And yes, stockpiling is usually done on site.  I know of mines where the tailings have been processed and reprocessed many times, sometimes for the same elements, sometimes for others.