That's a really interesting question.
I would have to say that, in general, no, things don't shift around. However, it's not that simple of an answer. Some games - by design - have black bars for UI reasons. Usually, however, it is on the bottom. And many times, while it seems like wasted space, it's often there for the occasional UI element to appear when needed (seems strange today for such a design choice to be made, but it must have been easier back then to simply allocate areas like that).
Here's an example - if you look at Loom
, scroll to the bottom where the screenshots are. You'll see many have a giant black bar down there, and at first glance it seems almost broken. And you go through the game for quite some time before you realize that space is actually reserved for other UI elements (e.g. see the last screenshot).
However, it is worth noting that in emulation I've seen shifting occur and I couldn't explain it. I only learned how to avoid it. Example: I loaded a game over the weekend in emulation and manually set it to NTSC. I knew it was a game targeted to N. America so set it up myself. However, on launch, I noticed over time that top of the screen was being cropped by about 10-20 pixels. Just enough to chop a very short slice off the top. I've seen this in 3 separate instances.
If I were on my A1200, I could reach behind the 1084S monitor and use one of the vertical control knobs to fix that problem very easily. The only way I could fix it in software was to tell the emulator to 'Auto' select PAL/NTSC and resolution for me. Ultimately it chose NTSC, but the cropping didn't occur. Makes no sense!
I've never seen any left/right anomalies, only vertical.
EDIT: I only made the OCS assumption because one of the retailers said it supported "ECS, AGA" among several other technologies and platforms. I assumed (incorrectly) that the omission of OCS was intentional.