Maintain stroke width for unproportional scaling

You have given us a cropped screenshot of the problem and that does not show us how the blinking eye was created. Mylemium thought it was a 3D layer. That is the same impression that I get. It could also be a 2 layer graphic with the anchor point right at the bottom of the top half that is scaled from 100, 100 to 100, -100.  From your reply to Millennium, that sounds accurate.

If you want to maintain the stroke width you need to add a stroke below a group, then transform the group. When the AI file was converted to shape layers groups were formed but it is very likely that the shape of the upper and lower eyelid is not a stroked path but a filled shape. For this to work, you need empty paths in a group that are scaled from 100, 100 to 100, -100 with the anchor point at the bottom of the upper lid, and a stroke applied after the group has been created. A screenshot will show you the setup a lot better than a description of the workflow. 

You will have to select the shape layer created from the AI file, press the U key twice to show how it was made, then start moving things around until you can get empty paths in a group with a round cap stroke applied to the group and an animated Group>Transform>Scale property to make it work.

Not sure what you're getting at. Apparently this is a 3D layer and there will always be some angle where the view axis coincides with the layer's orientation, resulting in a completely flat apparrance and the strokes being flat around that mark. You need to completely change your procedures and animate the actual shape paths in 2D if retaining a consistent stroke width is required. Some things can't be done quick & dirty with 3D layers.

Mylenium

If you show us the expression, we can help.

*Martin