My guess without an image - Perhaps they have aphids?

It could be easier suggesting what is occurring if you would post several images.

Helpful would be one of the plants & their surroundings, another a close view of an affected plant.

Floral, May I ask what your nasturium "bed" looks like throughout the year? It inspires me to try the same with my nasturiums, rather than to continue trying to train them up the fence around my vegetable garden, but I'd like to know what the area will look like when they're finished and how you cope with weeds. Thank you as always for your advice.

kitasei - in the Winter there is nothing there because they are on my allotment which is primarily a vegetable growing area. Nasturtiums collapse with the first frost. I either leave the vines to rot or cover them with whatever mulch I am using. Some years I even dig the area over but that doesn't stop them. They always drop enough seed to explode the next year germinating in late April/May. Once they are growing there are no weeds. I can claim no credit at all for their luxuriance. They do it all by themselves. In fact in summer I pull a lot up from places I don't want them or they'd cover the entire bed. In a previous tiny garden I did grow them up the walls and they just climbed up by themselves once I'd directed them with a few sticks.

great - I have just the place for them I think!

I happened to see a magnificent display of nasturtiums at the Maywood Gardens in Litchfield County, CT this weekend, open once a year as part of the Garden Conservancy Open Days. If you ever have a chance to visit this garden, do!