Instructions
Move through the glass doors which opened up right after you dropped the fossil on Dolph. Once inside, go up the stairs to the Springfield Museum and grab the "RoboBart" helmet.
Shoot the three targets with your helmet from left to right to keep Kearney moving. The last target on the right will lower the bridge and chase him into the second diorama.
Shoot the targets nearest Kearney to chase him to the right, and through the drawbridge towards the third diorama.
Continue shooting targets with your helmet in the following order: the man with the gun, the donkey, the soldier and then the man operating the cannon. This will chase Kearney out into the museum proper, where he is now vulnerable to normal attacks.