I checked the website for the Missouri Botanical Society to see their picture of the fine leaf buckthorn. See it here:
http://www.missouribotanicalgarden.org/PlantFinder/FullImageDisplay.aspx?documentid=15129
To my eye, the shrub has a pendulous nature rather than standing straight up. Be sure not to overwater; it requires good drainage. With the rainfall we have been having, it may not need supplemental water for awhile. Wet leaves will also cause the plant to bow and bend.
The plant can get a couple of different fungi, so check the leaves, including the undersides, for discoloration.
Thanks for using Ask an Expert.
 

Thanks, Sharon, let me show you what all the images show regarding its "posture"... mine just don't do that.

I notice the hose in your picture. Were the plants wet when you took this picture? Fine leaved and long stemmed plants tend to bend over when wet. (My iris are kissing the ground right now thanks to a short shower.)
I found the following information as I was doing more research. "Mature plants are drought-tolerant. Apply a slow-release fertilizer in early spring that’s formulated for shrubs, and follow all label directions." 
Wait a few days until the rains have passed through and let the plants dry. Also check to see if the ends of the branches show new growth. The plants look healthy in the pictures, but you might, when the buckthorn dries, take a sample branch with foliage into the Plant Clinic at the Extension 569 Hanley Rd., Central Point. Hours are 10am to 2pm, M-F. Plant material bags are available if you can't come in those hours and you will be called with their diagnosis.

A note from another Master Gardener in Jackson county: The "Fine Line" Buckthorn is a product of  crossing Rhamnus frangula 'Asplenifolia' with R.frangula 'Columnaris'. Many times, hybrids like this do not always behave as the photos and advertising show. Your plants appear healthy in your photos. it's quite possible that growth is moving away from the board fence--this is, after all, a shrub designed for upright growth in a pot, a featured spot in the garden, or a "hedgerow" which does not include being against a board fence where no light hits the back side of the plant and growth in that direction is discouraged.The recommended spacing is 36" to 60" apart, which can also be interpreted as needing that amount of space on all sides.  Another consideration is that this plant has been developed for cooler climates than ours. It does need well-drained soil and only medium moisture.I(see comments from Sharon May previously). You might attempt to stake the wayward plant that is growing outward, Use a stake in front and one in the rear and loosely "corral" the bush between the stakes with soft rope. You may have to start with a little bit of pressure and slowly increase it if the plant seems to respond. You may also just have to accept that this one's genes are a little confused!

Thanks, Sharon and Marjorie! This is shedding some additional light on this issue for sure. The fine lines were recommended for this specific situation next to the fence (I'm in zone 8a in Ashland) by a designer at Shooting Star Nursery. I understand now that the hardiness zone for this plant is 2-7. The spacing is 36", soil and moisture according to guidelines, and they get full sun from about 11AM to 3PM. The staking suggestion sounds great. There is no main trunk - the feathery look is due to many thin flexible "trunks", so not easy to rope or tape. Maybe I'll stop by at the Plant Clinic and get some further instructions :-) Thanks again, Marika