I filled in to help the CO change in our circuit once. Due to circumstances, the CO had to leave the circuit about 3 months early. I kept the circuit files and would help with the weekly transitions as substitute CO's came each week. The circuit file has a file for each congregation. The contents are mostly utility stuff (who was appointed as servants last few visits-etc) However, if a CO feels that a matter in a congregation may possibly be an issue again, the written letters associated are kept so that the future CO can see them and not start at square one all the time. The more prominent a brother was who became a problem and 'fell from approval', the more likely a letter 'since this man had been given great trust and is probably very capable, but we need to be cautious of him now for some time'. There is definitely a verbal exchange of the lowdown at the changing of the guard. Also, Assembly overseers and PO.s (coordinators now) are sought out for opinions and status, and I was always surprized at how much the comments of these 'heavy' brothers would be relied on.
Usually, the outgoing CO schedules the first few months of the new CO's route since he will be new to the area. The former CO usually assigns the assembly speaking parts for the first upcoming circuit assembly since he knows the brothers who are better speakers and or the 'best' examples.
District speaking assignments are made a different way since congregations possibly attend different District Conventions each year. A varying group of 3-4 brothers meet with the CO around January and decide who in the circuit would be qualified for various types of parts on a District Convention. The recommendations are forwarded to the branch, who then make assigments and send those to the speakers by direct mail. Even though they try to rotate some of the elders who attend this annual meeting, its still a good ol boys club. At this meeting, recommendations are also made for brothers who would be qualified to serve on appeal committees and the special committees.
I hope this helps with what you wanted to know