Use mouse on second screen

I am toying with making a multi-screen Android 11 device. I managed to start a different app on each screen, using am start -n --display . Also, I found a launcher, which is capable of starting apps on a screen of my choice. But only one of the screens is a touch screen, and I am trying to use a mouse for the other (or preferably both) screens. Sadly, the mouse is limited to the first screen, which is also the touch screen. So I have two input devices on the first screen, and none on the second one. How do I assign the mouse to the second screen? Is there any way to use the mouse on both screens?