When hovering a user's image and their flair shows up (ex: on any review page or on a question where someone other than the OP has most recently edited the question), the flair is requested after the image is hovered. It requests a larger sized image and the other flair information every time that the image is hovered.
If the image loads quickly (before the transition finishes), it starts out behind the rest of the object then pops to the front after the rest of the flair is done animating. Here's a GIF of the issue, though it's easier to see if you test it yourself.
Also, if the image loads after the transition finished, it won't be visible often times the flair will show and the image will pop on it once it's done being loaded.
This is, of course, dependent on the user's current connection and the response time of the server because it deals with the load time of the image.
This is true testing on the latest Firefox, Chrome, and Internet Explorer on Windows 8 across multiple computers.
This bug is still here (20 September 2015) after the profile redesign.