What could happen when you send a spy?
They may catch him on the border - tough luck.You will get no information.
The spy manages to get trough the customs and he infiltrates successfully, he can start sending reports. It will keep sending reports for 24 hours until called back or is caught by the enemy counter-intelligence.
Catching spies on the border is done automatically (you will know who sent it). To seek & destroy infiltrated spies you have to go to the counter-espionage menu and perform the busting yourself. From there you can see the level of the spy and if it is not more than two levels higher than your Border outposts, you will know the sender. Every infiltrated spy can be caught until it is called back home.
according to the manual, it only stated that activated spies will stay in the enemy's province for 24 hours. What about spies which has not been activated? How long will it stay at the enemies province?