I have a keyboard shortcut that lets me use macOS’s built in “Window >> Move to Built-in Retina Display” menu item. Unfortunately, not all apps have this menu item so it doesn’t work consistently.
I’ve noticed that applications that don’t have this menu item do still have the command available in the window’s zoom button popover:
I’ve experimented with using the built-in window management options to move windows between displays but didn’t find it as flexible as I would have liked.
Moom works well for this and many other window management activities. For example, I configured Moom to move the active window between my two displays, resizing it proportionally when I press ⌃⌘↩. It works like a charm!