My brass comes out of the chamber with around .001 runout. When I run it through the neck die (Redding, regular neck die or competition bushing type) the runout increases maybe another .001 or as much as .010. I have examined the dies and cannot pinpoint anything as the culprit. I do not neck turn, could this be the source of the crookedness? If not then what?

Try checking your shellholder for proper fit in the extractor groove. The case should sit in the shellholder with it's base squarely on
the shellholder with no rocking motion. I had a situation with some european brass which did not fit my RCBS#2. It was throwing them off by as much as .020!

Also, a technician at LEE told me if your neck sizing die is "tight" it causes too much tension when the expander ball is pulled thru and distorts the brass. The answer there is to neck ream or use a different die.

If this situation is constant among all your brass, this may not be an answer but sometimes if your brass has a heavy side or one side of the neck is thicker than the other it can cause a case to be somewhat bananna shaped when it is extracted from the gun.