This game was started with 16 civs on a standard map.
The golden globes are capitols. The red disks are cultural hotspots. 
We are playing Egypt (Yellow).

Three Egyptian towns (Yellow) are all we start with on this crowded map.

North are the Japanese (Red)
South are the Chinese (Pink)
West are the Germans (Blue)
Far West are the Americans (Light Blue)
Far South are the Babylonians (Dark Blue)

The cultural hotspot in this case is an Egyptian cultural attack on the Japanese. By placing a town close to the Japanese capital and by rushing cultural improvements, we can rob their town of resources. 

The Japanese declare war, and they that live by the sword were destroyed by the sword. Now, we must defend our captured towns from the cultural influence of the neighboring civilizations. At the cultural hotspot, rush-build Temple, Courthouse,  Marketplace, Library,  . . . . 

Egypt has consolidated its rule over eight towns. The battle to win the hearts and minds of the people is ongoing. 

The Americans start trouble. Egypt attacks, takes New York, and splits the American Empire in two pieces. The Americans sue for peace. Now the cultural battle is even more intense. Each contested city is closer to both enemy capitols than to Egypt's. However, the Americans are severely weakened. Most of their cities have no communication with their capitol. 

The Chinese declared war, and were subsequently eliminated from the continent by Great Leader Ramses with a Cavalry Army. Now, the German city on the inland sea is under cultural attack, as the Egyptian city to its northwest becomes a major cultural center in its own right. The German city is surrounded by Egyptian yellow.

The Egyptians declare war on the Americans and take out their capitol. The Americans reestablished their capitol in the far south. Being closer to the German capitol, the home of The Hanging Gardens, the western cities of the Egyptian Empire are now under cultural duress, Everything depends on the loyalty of one city. Sue for peace and consolidate.

The Americans insist on a fight. Egypt removes the Americans from the continent. The Egyptian city north of the German capitol is still the key. If it falls, then the entire Western Egyptian Empire would be cut off from the capitol. Build the Forbidden Palace. Or perhaps there's another way.

It is not a dishonor to declare war to protect your people. It is only a dishonor to say one thing and do another. Declare war before you attack. 

By removing the German capitol, we will solve almost all of our culture problems.

Taking a look at the whole continent, the cultural front line is now a recently conquered German city. It is close to the new German capitol, but more importantly, it is near the Indian capitol, which is very old and very venerable. Rush-building the Forbidden Palace in the hotspot may be a good idea, but only after rushing a few other city improvements first. You could lose your Forbidden Palace if the city should "flip."

Meanwhile, the new American capitol may eventually cause problems in the north. 

The two cities on the Eastern Peninsula flip to Egyptian control. The German capitol falls to Egyptian armor. (Note that the northern most Indian city is not in danger of flipping, as it is a very old and ethnically pure city.)

Now cultural tensions dominate. With careful use of rush-building, it should be possible to hold onto the position. Or declare war and conquer the Indian capitol,  cleaving their empire in two!