The game itself will have no solution since 90% and more wants to conquest the map with ease.
Even if migration is limited to 3 kingdoms, then people will all agree to move to the kingdom with least cities, and recapture the cities of the 1st,2nd and 3rd kingdoms once everyone moved.
We don't want precisely balanced factions.
This game is all about back and forth action, that doesn't happen when nobody has enough extra people to take the initiative.
Coordinated groups are also not going to separate to balance the game out, because once they do that they lose the ability to coordinate.
If you attempted to limit who can migrate where, even more people are just not gonna use the feature. Go to most of the servers available and ask yourself: "Would I ever have any reason to want to migrate to the bottom 3 kingdoms?"
Make it so every week the kingdom with the least amount of power gets 7 kings and no advisors according to the top 7 players in power in that weakest kingdom.
So 7 people move over for the insane benefits, everyone else doesn't because they'd get absolutely nothing for doing so? Doesn't seem to solve the problem or make it any more engaging.
How exactly are precisely 7 people going to make a difference in a server with hundreds of people?
Moreover this is just gonna involve 7 people getting screwed, because their collective power will make their faction too high on the power boards.
The simple reality is that the game will usually come down to a few factions. This is neither a problem nor something that needs solving.
The reason this happens has more to do with how the game works (particularly resources, where getting zeroed in the mid/late game is game ending because the rate of acquisition of resources without dominant map control is too low).