mm_flynn
absolutely correct from the UK perspective - I can't comment from the US one.
Over here, international flights are allocated an "ORCAM" squawk, one specific for the FIR you are departing to, and retained through the entire flight. All other flights don't have this luxury - every ATC unit has a set of squawks allocated to it, so every time you change unit you will have to change squawk.
The advantage of this is that we can see who you're working from your squawk. I believe for example that a 370x squawk means you're under Brize Zone control, while a 371x squawk means you're under a service from Brize radar - helpful as you know which button to press on the phone.