The system clearly prevents the gate agents from moving you from a BE seat to MCE. They get some kind of notice that makes it clear you're in a BE seat and are not allowed to switch.
Makes sense, although the first leg we were in MCE without asking for any type of seat assignment (auto-assignment). I think if the MCE seats are the last left, you can still get them with BE, but not by asking. I think the only thing you can gain by asking is changing the seats--up to their discretion of course.