I just had this happen with a BA ticket. The routing was SEA-LHR-FCO in coach with a 4 day layover in London. I decided to move the LHR-FCO segment out one day. Got it all rebooked and everything looked fine, except a closer look showed that they had booked the SEA-LHR segment of the morning flight (BA52) to London instead of the existing evening flights (BA48). That wasn't going to work as this reservation was linked to another with other people we were traveling with.

I called AS and asked for the partner desk and got the usual run around with how the first agent was "more than qualified" to help out. Well of course BA 48 had no more coach seats open so the agent eventually got flustered and proposed we just move everyone to BA 52. That wasn't going to work for everyone's schedule so eventually, grudgingly, she sent me to the partner desk.

The partner desk, of course, knew exactly what to do, since it was the fault of AS that things got messed up. He requested a seat from BA which came through in a couple days. Problem solved.