When it says to get X for population and economic, they mean be the first in your civ to reach that milestone. So when it changes, that means someone else reached it before you did.
Research and military medals go to the MVP (with the former going to whoever contributes the most to that tech, and the latter going to who contributes the most to the victory in battle). Culture is just randomly given, based on who contributed to the last wonder/wonder event finished, and construction goes to the first in the civ to build a given ginormous building.
