Out of the four airlines I've worked at in the last 7.5 years (SYX, ATA, AAY, SWA), all eventually didn't do "circling approaches." They all did "circling maneuvers," which is a visual traffic pattern in weather above 1000/3. I think this came about because the FAA felt that testing circling approaches in a sim were unrealistic or impractical.
And, from what I remember, none of the airlines I've been at had a specific procedure with the AP/FD (when equipped) regarding how a circling approach was to be flown that was any different than the standard PF/PNF duties. From what I can remember of our FOM at SWA, it's up to the PF how it's to be done (and, then, ultimately up to the Captain)... so there's no procedure, merely techniques.
And, from what I remember, none of the airlines I've been at had a specific procedure with the AP/FD (when equipped) regarding how a circling approach was to be flown that was any different than the standard PF/PNF duties. From what I can remember of our FOM at SWA, it's up to the PF how it's to be done (and, then, ultimately up to the Captain)... so there's no procedure, merely techniques.