Special effect in videos and gifs
For that specific example, any editing package would do - either an image stitching program or as Weylin said, more likely a video editing tool.

All that they have done there is have a background layer which is fixed throughout, then 5 layers containing the sign, his chin, his mouth, his tie and the ticker. 

Each of these layers is manually edited to move the item, then the frames are stitched together to form a gif.

There are ways to automate all the movements along animation tracks, but for a simple gif like this that would probably be overkill.