My questions is after a station or ship construction has been started, if the person who started the construction hops to a different faction. (one that is not a ship building faction, or a faction that has the dc for the station they started)
what will happen to the construction of the station/ship, will it continue or would it pause until the person is back in the ship building/ faction with the station dc?
Blot is correct. Provided there is an operator assigned, any operator, the entity will finish. On most occasions where a faction hopper is assigned the shipyard and NPCs will be automatically assigned (at completion) to the faction leader of the faction you used to start production.