After Effects

Not entirely clear on your description of the problem, but here's some thoughts based on your screengrabs:

First step is to get the design right.  I can't work out from your post how you've designed the layers, whether in Photoshop, Illustrator, or natively in AE, as everything is precomposed.  I'm guessing maybe in AE natively, as there don't seem to be footage items in the project window.  Personally I'd design in Photoshop or Illustrator, unless there's a specific reason, ie you need to animate shape layers.

You need to have every layer in the main correct position before you apply the Puppet tool.  Once you apply a Puppet mesh, if you scale or move the layer with any other tools you will probably disrupt the mesh and get weird results.  Don't duplicate layers that already have the Puppet tool applied.  Apply the Puppet tool individually to each layer where it's required.  If you duplicate a layer that is laready Puppeted, reset the Puppet tool and start over on that layer.

In your second image the eyeball is extending below the lower lash.  If you combine the lash and white eyeball into one layer, you should be able to distort them together with one iteration of the Puppet tool.

In your third image, the lower lash looks distorted and weird.  You need to lock the left and right corners of the lash to the upper lash corners, so the eye is always wrapped by the lash line.  Try using the Starch brush in the Puppet tool for that.

If I haven't addressed your problem, please repost with a clearer explanation of what the problem is.

in the future, to find the best place to post your message, use the list here, https://community.adobe.com/

p.s. i don't think the adobe website, and forums in particular, are easy to navigate, so don't spend a lot of time searching that forum list. do your best and we'll move the post (like this one has already been moved) if it helps you get responses.

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