How should gaps at edge of an asphalt driveway be filled?
First vacuum or blow out the leaves.

You can get "asphalt repair caulk" or "crack filler" in standard caulk gun tubes, which will work for the purpose as well as anything will.

If you are trying to level the offset (rather than just fill the crack at the joint) in the first two pictures, you'll need to move to "cold patch" to build the asphalt up to the level of the concrete.