I assume they used Illustrator files for that video, but you can do it from within AE. I think the easiest way would be to use shape layers. AE will create various standard shapes - rectangles, circles, stars, etc. - or you can hand draw shapes with the pen tool.
For most projects like this you would want to create a long thin comp and add a layer for each stroke. Then you animate the position of the layers, nest the lines pre-comp in your main comp and then bend, warp, or wrap your nested comp using distortion tools, perspective tools, or by masking and rotating 3D layers. Here's what that would look like. The screenshot is from a tutorial I'm working on and it should give you an idea of how to achieve this effect.
You can use any kind of distortion you want to bend your animated lines pre-comp. The Beatles Clip you referred to could easily have been done using this technique and 3D layers.
I'm looking forward to this tutorial, many thanks !!! really thank you