You might want to take a close look at the Puppet Pin tool.
There are also distortion mesh tools in After Effects under Effects>Distortion. There's also liquify. Depending on what you want to do Puppet Pin may be your best option. With any of the distortion tools in AE you can limit area effected by changing the size and or position of the points.
I'm guessing that you want to make the eye blink. I would probably create a 4 or 5 frame sequence and use mesh warp or liquify to animate a single blink, then nest the comp and use time-remapping to animate my blinks in the same way that you can use time remapping to do lip sync with mouth shapes.
The eye needs to be its own layer for proper distortion treatment. That's all there is to it. You really need to read up on basics.
Of course separate what you can do! I was wondering if it was possible to make from a single image ... commo the video you sent.
You can increase the number of rows and columns and then just animate the eye but the easiest thing to do with that image may be to add an adjustment layer that's just bigger than the eye and apply distortion mesh to the adjustment layer.
Most composites and animations in AE are easier with multiple layers. Trying to do everything on a single layer is not usually, if ever, an efficient workflow.
I will do everything separated by layers it is less work and more professional!
But just to learn more, tried to do what you mentioned. And I had this result:
When I change the duplicate layer as a mask that I created, I can still see the layer that is behind .... apparently this method not of right.
Actually, the adjustment layer idea won't work because I forgot that an mesh warp does not respect the layer size of an adjustment layer. It's a render order thing.
It looks like you are using Puppet Pin.. if so, you need the eyelid and surrounding area if you are trying to animate a blink. I would draw a rectangular mask around the eye and include plenty of extra area, then use puppet pin. If there is transparency in the eye then you'll have to fill that with something or bring the edges together.