From my understanding, all brands of dies use the same threads so you can use any die in any brand press.
I load 9MM and 45 Auto on my turret press. My 9MM dies are the Hornady 3-die set with a regular expanding die. When I set up for the turret press for 9MM I had to buy a powder-through expanding die to make the most of the press design. I also added a separate crimp die because I prefer to seat and crimp in 2 steps. My .45 Auto is the Lee 4-die set already had the powder-through expanding die when I bought it.
I haven't used a progressive press but I assume the steps (and required tooling) would be similar.