What preliminary work needs to be done before installing a window sill in the room?

We completely replaced our facade. Previously it consisted only of wood, thin wooden panels and wooden windows. Now new windows have been installed and the facade has been reinforced. The aluminum sill was put on the outside and covered with mortar, which is not even completely airtight. There is a lot of air going through here.

Now we wanted to install interior window sills made of stone.

Currently there is wooden batten over the stones and there is still bare stone visible under the windows, including the holes.

What precautions should be taken?

I think only foaming would be the wrong way to go. Should I glue insulation like XPS here? Can the window sill be glued afterwards or is more needed here? I want to avoid cold bridges and mold.