What does the XFR button on the Challenger 300's autopilot panel do?
You can find manuals for many aircraft including the Challenger 300 at Smartcockpit. The Automatic Flight Control System section includes the function of the XFR button.

There are two flight guidance computer cards that drive the autopilot, including the flight director and yaw damper. The XFR button switches how the computers are used.

For most of the flight, only one computer is use and provides output to both left and right cockpit displays. Pilots will get warnings if this computer fails, and they can push the XFR button to switch to the other one. Without a failure, pushing the XFR button switches so that each side of the cockpit gets input from a different computer.

Precision approach and go-around modes are the exception where by default each side of the cockpit is driven by a different computer.