You may notice that players enter the Multiplayer Setup in a different order than they did when you previously played. In this case, you'll notice that player A might be listed as player B.
i.e. the first time you started a 3 person game, it was like this:
Player1 -- Civ1
Player2 -- Civ2
Player3 -- Civ3
But, upon reloading, Players 2 and 3 entered the room in a different order. So, it's like this:
Player1 -- Civ1
Player3 -- Civ2
Player2 -- Civ3
You don't have to leave the Multiplayer Setup screen to get it right. Each player has a number to the left of the player's name in a square. The player will see a red square around their own square. They can click on it until the civ matches up properly. I think the host can do this for all players (but, I've never hosted).
You'll want it to end up looking like this:
Player1 -- Civ1
Player3 -- Civ3
Player2 -- Civ2
The same approach can be used to allow a player switch civs, or to take over an AI civ (i.e. if they get eliminated).