According to Cammarota, dustmen first became popularized onstage with with the play Life in London, W.T. Moncrieff's adaptation of Pierce Egan's popular novel about Tom and Jerry, two swells who get into trouble and have adventures together with the lower classes of society. The piece featured a dancing dustman named "Dusty Bob" who reappeared in a number of other plays as well.
Dusty Bob frequently caroused and danced with a black woman named African Sal. Together, they also had a mixed-race child named Mahogany Mary. You can see all three of them in this George Cruickshank illustration from the British Library:
Cammarota also brought in a copy of a poster advertising The Golden Dustman. Clearly, spectacle was important, as the poster announces (in much larger type than the name "Charles Dickens") "ENTIRELY NEW SCENERY by MR. JOHN JOHNSON". The poster goes on to list all of the locations displayed on stage:
Prologue--Sailor's Boarding-House near Wapping
Boffin's Bower in Holloway
A Street near the Temple, Fleet Street
Chambers of Mr. Eugene Wrayburn, Barrister-at-Law
A Church Yard near Temple Bar, by Moonlight
A Room in the House of Rumty Wilfer, near Battle Bridge
A Drawing-Room in Mr. Boffin's New House, in Portland Place
Another Apartment in Mr. Boffin's House
Exterior of Boffin's Bower by Moonlight
The Lock on the Thames at Sunset
Another Room in the House of Rumpty Wilfer
Drawing Room in Mr. Boffin's New House
The sets must have been a draw, given how much space is given to advertise them. (Far more space than is given to the actors.) Alas, I have been unable to find any images of what they looked like!