I have come to the conclusion that a comedian tossing around a faux baby is a British thing. I wrote about this subject twice before. Click here and here for those articles.
An unidentified British comedy film was described in the 1938 National Film Library Catalogue as follows: "A mischievous newsboy changes the babies in two prams outside a shop. A man comes out and wheels one pram away. A woman comes out to the other, discovers her loss, and chases the man. A policeman joins in the chase. The man is momentarily caught and in the altercation which follows. The policeman finds himself left with the baby. He throws it over a wall, but it bounces back into his arms."
This is similar to a sketch from the BBC comedy show Harry Hill's Fruit Fancies (1994).
A particularly outrageous example of baby doll abuse occurred in a sketch from another BBC series, The Big Train (1998).
This scene from Ernst Lubitsch's The Oyster Princess (1919) is similar to the famous baby doll scene from I Love Lucy ("Pregnant Women are Unpredictable," 1952).