There's a Black puppet character in my newest Polichinelle show. I have a pretty wide repertory company of stock Commedia-derived Polichinelle figures including Moricaud, a male Black servant who seems to be a clown but is actually rather shrewd.
One day he'll be used in his role of servant again, if I keep staging old Polichinelle plays, but for now I'm using him as Polichinelle's drinking buddy, filling in for Polichinelle's White friend and sometime cousin Niflanguille. Polichinelle still refers to him as his cousin in the show, which gets a small laugh, and calls him his "best friend" after causing him to be eaten by a crocodile, which gets a surprisingly big laugh. In the original old texts, Moricaud is meant to speak pidgin French, but I've dropped the dialect and no other reference is made to the character's color. He still suffers violence, just as the other characters do
As has been mentioned, the Jim Crow puppet and routine in Punch & Judy was inspired by a White minstrel performer's song and dance act, and so Jim Crow got into the show very much as Joey Grimaldi did, because he was a popular performer of the day. There were many minstrels, or Ethiopian bands, from which to draw inspiration then. Black servant characters and Jim Crow himself no longer relate to our world today, but foreign characters still do. In the US, policemen puppets often spoke like stage Irishmen, which doesn't seem offensive to me as my grandfather was a New York policeman who spoke with a very thick brogue.
Nowadays Indian doctors or telemarketers, Pakistani shop owners, Somali cab drivers, Mexican handymen, Russian doormen, Italian restauranteurs, bicycle-riding Chinese deliverymen, and Irish bartenders might appear in a show, should anyone want to put them there. They would reflect the greater community as it is now, including hoodies and rappers and customer service employees, as well as the surprise and even nervousness the community might feel about changes in its makeup.
A big part of the humor of Apu on "The Simpsons" comes from his accent. He also seems funny to audiences because so many quick stop market chains and gas stations in the US are owned and operated by Indians and Pakistanis, Hindus and Sikhs. Tastes in the US may have changed from the days when a turban-wearing Moor might appear in Punch and Judy and shout only, "Shallaballah," but Fred Greenspan has been able to keep an old puppet Moor in his show, and the Moor in his turban, by making him a magical wizard named Shallaballa who grants Punch three wishes.