when opening the necks of new Starling brass with New Dimension Custom Grade Dies, there are tiny bits of brass left on the base of the Hornady reloader. Is this normal? If not, any ideas what I'm doing wrong? It's new brass, has not been resized. Thanks.

If the necks don't show bad scratches, then probably not a problem.

New brass may have some burrs that are getting knocked off.  Have you trimmed / deburred the brass?  Might see if this makes a difference.

Personally, unless the cases were being badly scratched I wouldn't worry about it.  Seems like I always find bits of brass and other debris around the loading bench.