How does one get rid of the little runners from a Lilac bush?
You can cut down the existing suckers, then try weed barrier cloth covered with 3 to 4 inches of mulch, but there is no guarantee that will work.
Mowing will only exacerbate the suckering, not quell it. You can try Sucker Stopper RTU (Ready To Use), but it is very expensive. Right now a 22 oz. bottle is $33 online.
Here is some info. about Sucker Stopper:
Suckers come thru the weed barrier, they are very persistant. In some pruning books they suggest trimming suckers to ground level, removing the dead flowers and eliminating 1/3 of the oldest (read: biggest) branches per year. This will promote more new growth which produces the flowers and keep the shrub from only flowering at the very top. Hope this helps!