Since you want it out of the garden bed why worry if you kill off a section. Just pre dig a hole where you want it to be. Then dig up a section that is rooted and move it. Simple as that. You may have to cut it off from the rest of the plant, but again who cares, right? If it doesn’t make it try again with another section. Make sure you water it in real good.

My experience has been that clematis transplant very well, I moved my 3 year old clematis without setback this year, it blooms like nothing happened. But if I understand you correctly, the new baby plants are still attached to the mother., and getting nutrition from her...if they’re quite young I’d let them continue to root a bit more before I sever the ties. But if they’re already independently living, go ahead and transplant.

Layering - what you have described with fallen stems or branches rooting into the ground - is a favored way of propagating clematis. If the fallen stems have firmly rooted into the soil, just sever them off from the mother plant and transplant to where you want them. If you want them :-) If you don't want them, then the easiest approach is to just cut back the vine to a workable height and rip out all the rest of the topgrowth.

If you do move them, it can be done now if you wish or later. Wilting or dieback is always a possibility but doesn't necessarily mean the vine is dead in the root system. Just keep properly watered.

My experience is much the same as BenT's - clems are quite easily transplanted, depending on size. The older and more established the vine and the thicker the basal stems are, the bigger the rootball and the harder the process. Digging out an established evergreen clem (C. armandii) in a previous garden was a 3 day task and an excavation that was 6 feet across and 3-4' deep. And produced a rootball that weighed about 50 pounds!

Gardengal,

I’m interested in how you transplant such a large rootball intact? I’m always weary of transplanting an established shrub because the rootball fall apart and I lose so many roots. The plant will survive, but with significant setback.

LOL!! I didn't transplant that one - it was far too big to move!! I just needed it out of that spot to replace a fence and gate. But I have transplanted a number of other, less monstorous clems and any number of pretty good sized shrubs and trees as well. Just make sure the rootball is fully hydrated before digging. And it helps a lot to have more than one pair of hands if a mature, well established plant. Root pruning a good 6 months ahead of the anticipated move is also a good idea if you have the lead time. This will concentrate the important feeder roots closer to the root crown with less chance of major root disturbance.