Wanting to move multiple paths over multiple layers...

[maria_colls]

I am currently using Illustrator CS6.
My understanding of Illustrator is just above necessary at best. I have become competent at placing multiple straight lined paths to create mathematical driven patterns. I have primarily used Illustrator to create these graduated patterns as templates, so I can use these templates to cut out shapes. The link below is an example of my work, purely so you can see what I'm trying to explain. https://www.instagram.com/p/B2YuoyWpv1E/   As far as my personal relationship with my work, I would like to do to my work, like putting my finger into a part of the piece and pull down. Almost like it was clay and you were distorting a face.
I have tried using the warp tool, but it doesn't seem to work on multiple layers. I'm unaware also if I need to join the paths that would be affected. The reason that I want to have the layers underneath to move collectively, is so I finally can cut the layers, so the distortion.

[federico platon]

I am afraid, you may have to duplcate your work, the layers, and merged them to distort them.

But you may also use the Direct Selection or the Lasso Tool and use the Transform tools

[maria_colls]

Okay. I hear you. The reason it's complicated as the physical layers need to be separate, not merge, due to the fact the layers are also templates for me to cut each of the sheets out. That's why I included that image so you can see what I will do with each layer. Is there a way that I could merge the layers to create the effect, and then separate the layers so that I can use them to cut each of the layers on paper.

[Monika Gause]

It would be helpful if you could show your failed attempts. And also are we talking about a tool from the toolbox or an effect or something else? (Just asking because these days everyone calls everything a "tool")

[maria_colls]

Sure. I'm currently using CS6 version of Illustrator. I haven't been successful in making any changes so far. I have tried using the Free Transform tool. All that happened is that I moved the artwork across the screen. I'm at a loss what tool to use. I’m not too much how to display this file to you. Each layer is a template. I save each layer as a PDF and print it out, which is then used as a guide so I can cut the pattern out. The layers or sheets are then glued together. What I am hoping to achieve, is to create a distortion in the artwork but to create the distortion collectively to each layer, so that I can still cut each of the sheets out.

[Monika Gause]

What kind of distortions are talking about? What are the objects you create and how exactly are you expecting them to distort?

[federico platon]

Distortion Tools: Warp, Twirl, Pucker,.. tools work across several (visible or selected layers).

The outcome -stays- on its own layer. These tools are adjustable in size and intensity.

2. Also you may use a mesh to distort selected objects. Object>envelope Distort>Use mesh command. note this time distorted objects are grouped and placed in the very same layer. Save with other name, though it may not be reused for the same tasks, until you manually moved back each objet to its former layer. Use the menu:Select>Same... to acomplish this easily 

Enclosed find two images. 

[maria_colls]

Wow. I'm definitely going to have a go at this. I appreciate the time you took on this.