I have tried repeatedly with the built in TiledBackground object feature (sorry I wrongly referred to it as a behaviour) and have not had any success.
I appreciate the feedback. I have read the manual and run through the beginner tutorial (and several other tutorials, studied examples, and done lots of reading on the forums, etc.). I am simply not very good at this sort of thing, but I am trying (just struggling along the way!)
So, as a result of all that trying, I still have the problem of not being successful at adding a repeating textures to randomly drawn lines. I can get the texture graphic to display on the lines either stretched to fit, or simply one instance of the graphic to the correct scale - but sitting alone on a long line (not repeating).
Still experimenting/trying - no success so far.