Yes. I do this all the time in Altium.
Isolate the layer you desire with Shift-S then clicking on the layer tab you want ie: Mech1.
Select all objects on that layer using the mouse.
Copy - CNTL-C
Set the point of reference to the origin, for example.
Click left mouse button to set the copy origin reference point.
Then switch over to your new component editor window.
Select the desired mech layer ie:Mech1
Edit-->Paste special - Current layer option
Left click to drop it down.
Et Voila!
Alternatively, you could copy the entire component into a new footprint editor window then rip off all the layers and details that you don't want. Just isolate the layer (Shift-S), pick the desired layer (shift-mousewheel), select the stuff on that layer you want to delete with a selection box, then delete it with the Delete button. Repeat this until you are satisfied.