Q:How do I attach the NURBS planes, on which eyelashes grow, to the head so they follow the animation?
A: Duplicate the head and connect the duplicate to the main head using a Blend Shape with the World parameter. Next, to attach the upper eyelashes - delete all polygons from the duplicate head except for the 1-2 rows of polygons of the upper eyelid from which the lashes grow. Then wrap the eyelashes to these rows of polygons.
The issue is that your NURBS eyelash surface is being wrapped to the entire face, not just the edge of the eyelid, causing it to stretch during animation. You need to wrap it only to the eyelid's edge. Also, separate the upper and lower lashes if they are present.