In Boston, "I live down on Broadway" (not "down Broadway") would mean only that you live on the street named "Broadway."
I wonder if that would be different in New York, because New York has an Uptown and a Downtown.
Boston has a Downtown, but no Uptown. And in Boston, Downtown is not a direction. It is a specific shopping area in the center of the city.
"I'm going downtown to shop for shoes."
"Where did you buy that?" "Downtown."
In Boston, the old-timers used to say, "He lives down Maine" or "He's from down Maine." "From down Maine" is used only for Maine, and it's confusing because Maine is north of Boston.
But if you talked about the relatives you left behind "down home," you were from the Canadian Maritime Provinces -- "down home" to them but not to others.
If you were coming to Boston from New York, you were "coming on from New York."
If you were coming to Boston from anywhere else, there's no special word for it because it's only to be expected that if you are from somewhere else, you would naturally want to come to Boston.