What is the origin of the "...on the X" phrase?

Example phrases:

• "...on the regular"
• "...on the daily"
• "...on the down low"
• "...on the DL"

I'm curious whether these have a common origin. (Maybe there are more examples I can't think of right now too.) Thanks!

On (prep.)

Used as a function word to indicate manner of doing something —often used with the

on the sly
keep everything on the up-and-up m-w

11 a. In (a certain way);
Also forming adverbial phrases of manner — on the cheap, on the sly (OED)

Other examples that come to mind: on the fly, on the double, on the rebound, on the q.t. Of course each phrase has its own etymology, but I think this explains the function of on. The OED's citations for the "in a certain way" meaning go back to the origins of English.