This strategy is normally known as a douche – you delete your first towncenter and build it on your opponent’s face.
Typically persians are a favorite for this due to their town centers having double the hp (and consequently half as expensive to repair).
Any civ can do it but they are really the only ones with a true advantage.
For defending against it, its generally best to not repair your tc (it is very expensive to repair a tc), or try to fight back and just collect enough resources to make a new town center right after yours goes down.
If they drop their tc on you again you just repeat.
Their economy is behind yours because they had to walk all their vils across the map and also garrison them in a tc for an extended period of time so if you can hold and stay even then you should be winning.
Also, delete any farms you have that are under their new tc so they cant steal them.
One side note, it is worth it to repair your tc if you have clicked feudal, and only til feudal completes.