I'm currently running Ubuntu 15 desktop with an ATI R9 390 and a 4k monitor via display port. Everything runs smoothly. I have a second monitor which is VGA-only, and since the R9 has no VGA ports, I've added a Radeon HD 5770, and plugged in the second screen there.
Ubuntu detects the new GPU and screen, and loads everything fine. However, I've got a considerable amount of lag and poor performance when this second card is plugged in, just on the desktop. There are many small screen freezes, especially when moving the cursor over on to the VGA screen. Moving windows around causes freezes. Any kind of animated menu and such causes small freezes and lag. Also the HD 5770 is running it's fan like I'm playing a full screen game, just when showing the desktop with no programs running.
I've tried this on Mint 17.3 Cinnamon, Mint 17.3 Mate, and Ubuntu 15. Results are the same. I've tried this with a GeForce 210 (much less powerful card) instead of the 5770, and the problem is the same but much more pronounced. I've tried greatly lowering the resolution, and the problem is much less pronounced.
On Windows, this problem is not present at all.
It feels like when I plug in the second less powerful card, Linux is using it to render this large 2 screen desktop, rather than putting the work load on to the R9, like Windows seems to do, though I'm not sure of this.
edit: forgot to mention, I've also tried every driver that's shown as available in each of these operating systems. They either don't effect the problem, or they stop X from even starting.