OP: I don't see why not. There might be a slight dimensional differences. As long as the round chambers and you get good neck tension with no setback, it should work.
One concern. When I bought my .38/357 dies, I asked the guy if I could load both calibers with the same dies. He said that you can use .38 special dies to load both. But if you bought dies specifically made for .357, you couldn't load .38 special. It was a problem of either crimping or flaring, I don't remember which. Basically, the shorter .38 special case might not be able to reach in a .357 specific die. (I don't think anyone makes .357-specific dies, anymore, anyway).
So you might be able to load 9mm with .380 dies, but not the other way around. Wouldn't hurt to try.
Also, because the 9mm is tapered, it might be the case that 9mm dies are made with the sizer ring higher up. So you might not be able to size a .380 case all the way. And conversely, using a .380 sizing die on a 9mm case might overwork the web, unless you leave a gap between the die and the shellholder.