Thanks for the file, Craig. I never have had problems with this set up—if you like to share some details, I look into it.
Your file was set up to Projection>Frontal, which is kind of a dangerous set up. It is independent from any camera position, always the same location in the image-frame, the same movement as well, if the texture coordinates are animated. Again: This is a rare setting to use, as it ignores even any object movement as well, not to be confused with “Flat”.
I have set up the scene how I understood your notes.
The “NULL” under the object is not really needed, but I use it to remind me that the TEXTURE TAG uses only “relative positions”. With this set up the POS_ROT null [red circle] will allow to produce as well “relative coordinates”.
I move the Texture Space (cylinder—in this case) along a spline [blue] and a spline.rail [green] to have absolute control. This moves the yellow noise line exactly as I want it.
Note that the “Tile” is off for this. (I left your material in the Object Manager, but switched it Off: X). The smaller the texture space is, the better the resolution, except when you use a shader of course, which are most of the time resolution independent.
To answer your question about UV[W], I think that it is just fine, no concerns here.
[I switched the Reflectance off for the neurons-backdrop—just for demo purposes, to see the effect in a better way, please switch it on when you have explored the set up]
All the best