I know a lot more about BA than EK, but what I know about EK is very shortly stated: No seat pre-allocations in economy (or at least in discount economy). At least that was my experience in November 2002.
On BA, about 30% of the economy cabin is available for pre-allocation. This is done in bands across the cabin, rather than by numerical count. (For example, all of rows 25, 26, 27 and 28, including the dud middle seats.)
Once those seats are gone, they're gone - and the rest of the cabin is then only normally available for pre-allocation at check-in. The exceptions are for higher tier frequent flyers (BA and other oneworld) whose requests can be sent to the seating department who can override and pre-allocate anything else in the cabin.
If you can't get a pre-allocated seat to your liking, ask res to put a request into your booking for what you'd prefer. You never know if anyone's ever going to read it or action it, but it can't hurt.
If you're a BA Executive Club member (of whatever tier) you can log in to the BA website and check-in at -24 hours. Anyone else can register on the BA website and check-in at -12 hours.
If you're not already a BAEC member, you can now join only if you have flown on, or have a current booking for, a premium class ticket (I think World Traveller Plus counts as premium for these purposes) or a flexible economy ticket. There are said to be some obscure ways around this but I wouldn't count on it.
But online check-in is not infallible. Often you cannot change your seat even though check-in has already been opened - you simply get what you are given. On most flights, the seat change function comes live a little while later (often somewhere between -15 hours and -12 hours), but I have had flights where it hasn't come live at all.
If that doesn't work for you, go to the airport early. If you can use a self-service machine, you can see for yourself what seats are available for you to move to, if you don't like what you first get. Although the entire flight might still be locked up, at least you know the machine isn't being "creative" in what it tells you about what's available and what's not.