I have never had my poppies run. Rarely do I get any seedlings. Our soil is clay and things tend to rot, even though I have added a lot of compost through the years.
I have transplanted oriental poppies, and had them stop blooming for two years. I had an old plant (15 years) that I moved that did not bloom the following year, but the area where I dug it up, produced a new plant and that one bloomed a lot that year.