I'm looking for a utility that moves app windows between multiple monitors in KDE.
KDE supports it (at least in version 4.8), but the relevant shortcuts are disabled by default.
To enable them, open
systemsettings and browse to Common Appearance and Behavior -> Shortcuts and Gestures -> Global Keyboard Shortcuts. Select
KWin under "KDE component", and search for
window to. You will see several "Window to Screen n" and "Window to Next Screen" actions. Since you probably only have a 2 or 3 screens, it's probably simplest to create a shortcut for "Window to Next Screen" only (e.g. Win+space).
In even more recent KDE versions (System Settings 5.14.5, KDE 5.53.0):
System Settings →
Window to next screen (or similar)
The functionality works extremely well and can also be used to move full-screen presentation windows to the correct screen if an application goes full screen on the wrong output.