If the UK has exited the EU by next summer....
 

If the UK leaves the EU, how does that affect flights?

For example, we arrive in Dublin on American. We have a flight booked separately to London on British Airways. We go through passport control in Dublin, of course. Do we go again in London because now that's an international flight, or is that still considered a domestic flight?

From what I've read, the pull-out will take several years.

The issue here is not the EU. It's the Schengen Area, the 26 European countries that have eliminated passport control for travel among them. Neither UK nor Ireland have ever participated.

Now that I think of it, I assume it's the reason there's passport control before boarding Eurostar trains to/from London/Paris.

Dublin to London has never been a domestic flight. Travel between the Common Travel Areas do not need a passport & that's why there is no passport control. Brexit is expected to take several years.

The Common Travel Area only applies to British and Irish citizens. There is no right of passport free travel for anyone else.

Ireland does check passports on arrival at airports.

Everyone else needs passports (and visas if needed) or ID cards for EEA citizens.

Maybe my memory is playing tricks on me, but I don't remember going through passport control or customs when I was flying from Shannon to London-Heathrow a few years ago. Obviously, I would have been able to show my ID card, but I don't think that there was anyone who wanted to see it.