Momentum is conserved in the collision between the big rolling ball and the smaller rolling ball. The correct response would be:
The big ball transfers some of its momentum to the small ball, but it keeps going forward, too.
In a collision, the total momentum before the collision equals the total momentum after the collision, assuming no external forces are acting on the system. The big ball, upon colliding with the smaller ball, transfers some of its momentum to the small ball, which causes the small ball to move. However, the big ball retains some of its momentum, allowing it to continue moving forward. Thus, both balls end up with some momentum after the collision, preserving the overall momentum of the system.