You can use the Pose Scale controls to animate the Z scale of the object.
- Create the first Pose at time 0 seconds.
- Create a second Pose at time 5 seconds.
- Select the second Pose and open the Advanced Animation Controls.
- Change the Local Scale *Z* value to something over 100 (try 300 or 500).
- Play back the animation. You'll see the scale animate over the Transition.
You'll then want to edit the animation to your taste.