With us, we alternate legs. This is discussed, i.e. the captain states during preflight "why don't you fly the spoke legs (many different airports-and no VNAV/LNAV!! [most in Ppruneland are thinking-'oh my goodness!' or 'oh s--t!'] , since most FOs don't normally get to do that many airports (versus the same three hubs-various arrivals with restrictions, but almost always the same ILS approaches and runways to land on) ? All of our FOs by now have 6,000-15,000 hours+, in propjets and/ or only turbofans.
On the runway, as we wait for takeoff clearance, I say "you have the airplane-want the parking brakes set?"
Enroute, we normally never give the plane to the other guy/gal unless we need to go to the WC (lav.). BUT-we train, as captains, to give the plane to the other pilot in case we need to look at the "Red-bordered (EMER.) checklist" or in the COM (QRH) for an abnormal procedure, or even discuss a passenger medical problem with cabin crew-
For example, one time an FAA inspector was on the jumpseat as we climbed south from STL and the "Fire det. loop" light came on (!). This could indicate a major bleed air leak or an actual engine fire. I said "eh-why don't You fly the plane, maybe you wanna use the autopilot too, and tell departure we need to level off for a short while?" He is flying solo. just a short while......I hope. After finishing THAT ok (just a detection fault) and putting it in maint. logbook/sending description to Maint.C via ACARS, said , "ok, I'll steal the plane back from you".
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With any big problem 1) Fly the plane [who flies?], 2) Identify the Abnormal (not always simple) 3) Read/Do the checklist. ........Do not hurry.............