dig out as many as possible and if you decide to use weedkiller then Ive found using a paint brush to paint it onto the leaves works well. You can also hope that the poison travels down the runners too. In the past I have also used an old kitchen knife to dig them out the grass.

When it gets to the stage that there's more cinquefoil than plants in your border, most perennials can tolerate being dug up to have the roots washed and any bits of weed root picked out, then you could park them in pots while you massacre the border.
I think timing is important though, eg oriental poppies won't like you interfering with their taproot while they're just coming up to flower.

Thanks for your help, I think we'll have to tackle the lawn with a selective weedkiller first, and I was wanting to dig up some clumps of perennials anyway, so maybe a good clear out is on the cards, as you suggest, Teadrinker. They certainly are deceiving, these weeds, cos they look so little in leaf size but when you follow a runner to its root, they go down very deeply and are devils to dig up without breaking!