As long as all of the subject bird stays in the shot for its entire duration, you should be able to make very good use of the Pan Behind tool.
I would duplicate the layer then use the standard AE tracker to stabilize the top copy of the footage by picking something very near the bird to use as the TrackPoint. Then I would create a circle shape layer above the stabilized footage and use that as a track matte. You could then scale up stabilized layer.
Sounds like a great solution.
I have now stabilized the footage - how do I make the cirkle, so only the bird is showen? In AE og PP?
Have you had a chance to play with the "Magnify" effect (Effects > Distort > Magnify)?
It creates a circle by default, but the Shape parameter can be changed to a Square. When using this effect, I find it helpful to apply it to an Adjustment Layer that's stacked above the layers that appear inside the "magnifying glass".
Use the shape tools and pick the elipse. This will create a new layer. Press F4 to change the switches and modes column to reveal track matte and set the shape layer as an Alpha track matte.
If you still do not understand them type "shape layer" and track matte in the search help field at the top right corner of AE and study up.
First in Premiere Pro, I cutted the part out, where I'd likke the magnifiyng glass should be - and copyed this in the V-line above. Now I rightclick this new copy and clicked "Replace with AE compo..."
Now in AE I started with "Tracker" -> "Stabilize Motion". Then I added a layer the shape layer and made a circle.
How do I make this circle follow the tracked motion, and how do I make the circle be in the same spot during the entire clip?