Above is a clipping path consisting of text converted to paths and two rectangles. There are two colored rectangles pasted inside for the blocks of color you see.

This is a shape that I'll be using as the custom Envelope. This will be applied through the Envelope toolbar. First I Copy this shape.

Then I select the clipping path and in the Envelope Toolbar, I click the button Paste As Envelope.

As you can see above, the paste insides did not warp to the custom Envelope. This can be a problem if you do want those objects to warp with the rest of the shapes. Below is how to deal with this shortcoming.

I go back to my original clipping path and first Cut Contents to get those shapes out. Then to see my clipping path warping better, I send those shapes to back.

Select both the clipping path and the other shapes, then Group them all together.

I then apply the custom Envelope as described above. As you can see above, the other shapes are nicely warped.

While the Enveloped objects are selected, I click on the Release button in the Envelope Toolbar. This releases all the objects with the warping still applied.

Note: The Envelope is no longer live and editable after releasing the objects.

I then Ungroup the objects. Select the objects the are to be pasted inside and Cut them. Then I select the clipping path and Paste Inside. As you can see above, everything looks great!

The same problem happens with the Perspective Grid as well. As you can see above, anything pasted inside, doesn't perspectify like the rest of the objects.

Following the same procedure as described for Envelopes above, you can perspectify the contents of clipping paths too.

As shown above, I grouped the contents with the clipping path then applied them to the Perspective Grid.

Once I was happy with the Perspective projection, I went to the View menu -> Perspective Grid -> Release With Perspective.

Then I Ungroup the objects, Cut the rectangles and pasted them inside the clipping path. There... all nice and perspectified.