I know hollies sometimes propagate themselves when lower branches rest on the ground, but I have never heard of runners being a significant problem with them before. I don't think there is much you can do other than to sever and remove the upstarts. The earlier you can do this, of course, the easier it will be.
@Douglas Hunt Thank you. It is already too late to get an early start. When the weather cools a little I will get serious about removing them.