This should be posted as a sticky at the top of the forum.
Default behavior for E-mount is to set the lens to selected aperture immediately in M or A mode, and keep it there. In P or S mode, it will tend to stay open.
When focusing, if previously stopped down, the lens may open back up a bit to acquire focus, then it will stop back down again to meter, and maybe to fine-tune focus.
If Setting Effect is off, the lens will behave the same in A and M modes as it does in P and S modes - that is, it will stay wide open and open stop back down after focus is acquired.
All of the above is for AF-S, and with most E-mount lenses.
In AF-C, the lens will stay stopped down while the shutter button is held or half-pressed... so shot-to-shot focusing in continuous shooting is done stopped down. However, if the shutter button is released, focus acquisition the next time it is pressed is the same as in AF-S.
For a few lenses, like the 24-70GM, the lens stops down immediately even if setting effect is on. Sony has supposedly done this to mitigate focus shift issues seen with these lenses.
And, yes, for the most part, this is something that EVF/mirrorless cameras do differently than OVF cameras... light gain through the EVF makes composition and focusing wide open mostly unneccesary. Metering stopped down is more accurate, and composing stopped down allows continuous display of actual depth of field with no worries about focus shift when you take the shot.
The main complaint about this behavior that I have seen here is from people that shoot at small apertures in the studio, using lenses with the focus shift issues, and have a workflow where they keep the shutter button half-pressed (or BBF button pressed) and use AF-C to follow their subjects. It doesn't take as much light to track as it does to initially acquire focus, but, if your subject is moving around a lot, and the ambient light is low, it can be hard to keep a lot with the lens stopped down.