It's probably a skinning issue, or misplaced eyes/head bones.
Generally you should make base Head and Eye Look morphs in CC with respective bones.
If desired, an additional mesh morphing could be added for Eye Look Morphs.
I made this routine for the character straight out of AccuRig.
The only issue in the latest version of CC with Head morphs is,
that after making every Head morph you should restart Face Editor (weird issue and maybe related to my CC installation).
And in Preferences > Face Profile Editor, the setting should be "Remove All"
Here is that character in Live Face
77% of original size (was 657x19) - Click to enlarge