For US Travelers traveling to Ireland/UK and Schengen Area (Multiple entries)

This doesn't really answer your question of the process between Spain and Dublin; but once you are in Dublin ready to return to the US you actually go through US preclearance immigration in the Dublin airport so that when you arrive back in the US it's like getting off the airplane on a domestic flight. (Assuming no stops between Dublin and US outside of the US.)

I've done it often. I'm trying to figure out your question.

Did I have a problem? No.

When you arrive in Ireland, you go through immigration and customs.

When you arrive in London, thanks to the CTA, there is no immigration.

When you arrive in Spain you will go through immigration (Entry into Schengen).

when you exit Spain, you will go through exit immigration (leaving Schengen.)

Upon arrival into Ireland, you will go through immigration.

Upon exiting Ireland, you will go through pre-clearance US immigration.

Each immigration procedure is different depening on which agent you speak to. You shouldn't have a problem. Just answer any questions they may ask.

What Frank said.

Just a note that one clears US immigration in Dublin or Shannon only if the flight from Ireland lands first in the US. If one flies Dublin to Toronto en route to home, for example, one clears immigration in Toronto. If one flies Dublin to Reykjavik before landing home in Boston, one clears immigration in Boston.

When you arrive in Spain you will go through immigration (Entry into
Schengen).

And customs.

Thank you for your thoughtful responses. To clarify, if I enter Ireland with my US passport, will I be able to travel to London or Belfast without having to go through immigration again, thanks to the CTA agreement?

I appreciate your help!

To clarify, if I enter Ireland with my US passport, will I be able to travel to London or Belfast without having to go through immigration again, thanks to the CTA agreement?

Generally yes, though it's always possible to be asked to supply your passport. Several people on our train from Berlin to Copenhagen were asked to display their passports at the entry to Denmark.