It is a matter of checklist philosophy and procedures. At 'my old house' we went with a procedure that after the immediate items, the F/O took the airplane and was PF. The Captain then ran the checklist.
We then get to 'verify' and 'confirm'. It's been a while but IIRC 'verify' was after an action (verify altimeter 29.94) and 'confirm' was before an action was taken (confirm number 2 fire switch).
Our checklist procedures evolved over a period of time and came out of the kung fu drills that occurred over the throttle quadrant with emergency descents if the F/O was flying. The F/O was reaching for the speedbrake while the Capt was adjusting his glasses to try and figure out the pressurization panel. Solution? Capt assumes flying duties. Once we did that, we looked at who did what and when.
Since the Capt is the one making decisions, we gave him the checklist, told the F/O to put the airplane on autopilot and let the F/O program the autopilot. It works VERY well.
So, who pulls the switch? At our house it was the Capt but that was due to the checklist design.