I am trying to bring shrines (and mines) into the save guard area of a city, because then they can not be damaged by opponents that are not willing to pick a fight with my city.

If a finished shrine or mine connects to a city, the city adds the shrine to the list of things needing to be built (and essentially starts from a scratch status, using up a load of turns of the cities production.

So a working shrine requires to be "built" again as a building within citylimits?

Edit: [additional Note: After a few rounds the shrine indicated to be building for 10-20 rounds was recorded as finished?)

This might be a more general inconsistency as shrines and mines in the open field build (after being paid for and all in parallel) without using up the production of the city and are finished after 5 to 20 turns depending on the type of building not on the cities production.

Suggestion:

I think a shrine or mine should always require city production resources (currently not applied for open field construction) and on the other hand there should be no build effort required when a shrine or mine moves to within the citylimits and therefore into the protective arms of the city troops.