I know quite a few people that would be grateful if their Russian vine was sick, they are totally rampant and indestructible. Normally. In fact just one would be enough to submerge the average suburban garden, let alone four! Unless your pergola is on a par with the Arc de Triomphe. See, they've already travelled to the top of the pergola in only a month - imagine what they will be like in a year, ten years. And weedkiller won't kill them, I'm told.
I think you should let a few of them die and plant a nice honeysuckle, jasmine and clematis instead.

If you ever findout what is doing it please,please,please market it you will make a fortune.

I can't begin to imagine why you have 4 of this dreadful plant,which in my opinion should be banned from sale.The pervious owners of this house planted it and then trimmed it and threw the trimmings into the field next door. It rooted,and roots where it touches,it's smothering and killing full grown trees. In 4 years I have spent a fortune on weedkillers (most don't touch it), finally resorting to the evil stuff they use on Japanese Knotweed, spraying it and injecting it directly into the stems, am I winning- no, don't think so.

Just as a footnote, Russian Vine doesn't set viable seed in the UK,nor does it's cousin Japanese Knotweed, but recently hybrid seedlings have been found - deep joy.

Poor Amips, what negative comments on the Russian Vine which has its uses though mainly for covering up eyesores quickly. I think your problem is dryness at the roots and depending on the size of the pergola and the proximity of the vines to each other they may be competing for water. Another thing that happens is that as they age the leaves on the top shade out the leaves below and it is normal and natural for the lower leaves to die eventually leaving you with a tree like twisting trunk. I would seriously take the other's advice. Its so tempting to grow something fast growing just to get a bit of greenery but believe me in time you will regret planting 4 Russian Vines or even one. And if the Pergola is within 8 foot of your neighbours it will soon ramble into thier gardens too and you will be faced with hacking it back every month or so or facing the stoney looks of hostility from your neighbours. I would advice you to dig them up, prepare the soil and grow a mixture of fast growing reambling roses, clematis montana and a honeysuckle if you want instant coverage that is more manageable. Good luck!

Sorry, Amips, I was meaning to be jokey rather than negative. Still mean it about the vines, though. I think they can look good as a late summer plant growing unfettered over a chain-link fence, say. Well away from any private gardens, though. A big pergola covered with a frothing mass of it could look really amazing - just a danger to the horticultural public!