Can you move population to new cities with settlers?

Solution 1:

No, you can not move population with Settlers.

Settlers and Workers in Civilization 5:

• Do not (directly) reduce population in the city they are built in.
• Can be built by cities of any size, notably without destroying the city.

In Civilization 4, (possibly earlier, but I don't remember) those limitations were replaced by workers/settlers using all excess food in the city while they were being constructed, effectively stopping growth.

In Civilization 5, settlers still use all excess food, stopping growth. Workers do not use food and can be built without any special limitations compared to other units.

To the best of my knowledge, if you did build a settler anyway, you can't add the settler to an existing city, but only start a new city.