In my experience, alex, it does stay on the bottom insert part of the pan. That may be why quite a few tart pans are sold. Folks have to buy new ones because they never got the bottom part returned after serving the tart at a get together, etc.
However, if you want to remove the tart from the bottom, you could line it with a piece of parchment paper before you add your tart ingredients. Then, you could slip a spatula or knife between the parchment and the bottom of the pan to remove the tart.