The consequences should depend on the stage of the game you are at, and your government type. For example, in a modern era, technological advances allow governments to continue regardless of physical location, in earlier times this is not the case. The business of government required everything to be physically in the one location.
Perhaps some sort of government unit would be spawned when a capital is taken, that could settle in a new city and create a new capital.
Something to adress the 'nation' issue should also be done. The game seems based on the idea that nations have a common sense of identity and allegiance from the earliest days. This is not the case in the real world, and an element of this could be worked into the game. A more city state type development in early stages, empires that rise and fall.
These are only possibilities. Feedbackk welcome.