Several issues with moving a fullscreen game between multiple displays
Observed with a two-display setup (a laptop and an extra monitor), using Win+Shift+arrow keys. These issues are not 100% consistent, but can seemingly pop up or disappear upon toggling fullscreen or switching renderers.
If the game is in fullscreen OpenGL, and present on the primary display, attempting to move it to another monitor will cause a static "image" to be rendered to the second display, while continuing to render the game on the primary display. If the game is running in fullscreen OpenGL on the second display, moving it around works without any problems.
Not strictly related to multiple displays, but sometimes switching from OpenGL to Software will cause game drawing to halt entirely, most commonly on the primary display. Moving SRB2 to the second display will make it draw fine on that display, but moving it back will freeze it again.
If SRB2 is not in fullscreen mode, but otherwise matches the screen resolution, it will revert to the primary display. Attempting to move the game to another monitor will cause it to not be displayed at all, until it is moved back to the primary display.