Ya know, it just depends.................
I took a *A RTW in 2001 ex Canada. (In fact it was on 9/15!) The ticket's first trans-oceanic segment was supposedly on TG from LAX-NRT. On 9/12 I called to reserve a YVR-SIN seat in A on SQ. Just stated I had an "open segment". When I actually showed up at the counter, I just told them I "had been rerouted due to 9/11". I just got my boarding pass. Trip was great. In Bali I decided to change itineraries since I wanted to stop in IST. Called the local Bali SQ office. They rebooked me. At the airport I had to go to the ticketing area. It took some time but they reissued the whole ticket for the fee--it was either 75 or 100 USD.
Eventually I stopped in Germany at my parents. Had planned to explore Europe, then do NYC, PHX, LAX before going back home to SFO. By the time I was in Germany I had spent enough time in hotels. Although I used SQ for most of the trip and wanted to do the FRA-JFK flight on SQ, by the end of October, the 9/11 stuff hit me and I just didn't want to go to NYC. I didn't want to fly UA JFK-SFO. I decided to skip the rest of the trip and go home directly on LH FRA-SFO.
I went to a LH City Ticket office; they couldn't help me. I then went to the airport in FRA. Despite going to the SEN desk, it was a nightmare just to get a simple segment. I went to several different counters until someone agreed to help me. Once someone did, it took 45 minutes!
Bottom line: it really doesn't matter on the airline; it depends on the individual agent.