You do not have to do inputs after you defeat them, rather you need to meet some prerequisites:
(a) Play Medium or higher difficulty (b) You can lose a round but not a match, i.e. no retry/continues (c) Does not matter if you transform Shang Tsung into a baby when he is morphed into other characters (d) Do not successfully block the opponent's attack in the deciding round, successful blocking negates Babalities (e) Perform Babalities/Fatalities on all opponents, there is a minimum number of Babalities that needs to be done, but I have yet to determine it.
Anyone who has time to determine the minimum number for the last point, please let me know.
I came to that conclusion because I have done a stage fatality on the first ladder and subsequent Babalities on the proceeding ladders and still manage to perform the Babality on Kintaro/Goro
Spoilers (Kintaro/Goro Babalities described)
He turns into a kitten (?) and drinks milk from a dish
He plays/claps with his four hands and accidentally slaps himself and cries