I do not think that there's a direct way of doing this, but here's a workaround:
Bring the gradient AI layer in AE.
Take a screen shot with the gradient Mesh selected to see the control points and bring that as a reference in AE as well matching it with the AI gradient layer you already have imported.
Create a solid- comp size- precomp that and apply the Mesh Warp to the precomp. Turn off the solid layer in the precomp. You will see the screen grab with the gradient mesh control points under. Set your Mesh Warp to the same number of columns and rows and move the distortion points to match the points on the gradient mesh.
Bring in the gradient layer from AI in the precomp. Apply Mesh Warp to it. Turn opacity to 50%. Open the precomp and the main comp in dual view and adjust back the gradient in the precomp to make the distorted image in the main comp match the gradient underneath. Basically you have to move each point in the exact opposite direction in both x and y.
Hope that makes sense to you.
Not the easiest way though- maybe someone else has a better way.