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Expansion should in my opinion bound to milestones allowing you to claim more regions as "core". 

I really like that idea. The UI might be a bit tricky, but I think it's worth an attempt.

I think you'd need more systems in place, like happiness or diseases, before an adequate soft-cap can really be implemented.

That's one of the plans for settlement happiness. Settlement size would decrease happiness. As settlements grow larger and larger you'll have to acquire more and more luxuries to keep the population satisfied. Diseases would be another good (and historically accurate) way to soft cap populations.

>The UI might be a bit tricky

We'll need a region view/overlay in any case, to see where the regions end and who owns what. It could be done through there in a fashion similar to building upgrades ("influence area of settlement A" "1: claim as core" "you need X citizens to claim as core"). Or, if we go by my favoured idea of Town Hall buildings, inspecting a town hall (or trying to build one) may tell you what you need. The Settlement Info screen might also see use here, with different pages. I think eventually this screen is likely to require some expansion anyway as a menu or ledger of sorts, to see everything that's going on with the settlement.

I really like the idea of certain structure providing administrative functions in general. Going back to what I said earlier about not selecting a city like you do currently, you could move most or all of the settlement info menu to the town hall. Surplus management is very important, but not until a certain point in a settlement's development. Or perhaps add a quartermaster for surplus management and have the town hall for planning functions like the top third of the settlement info window plus info necessary for expansion, etc. This could also provide ways for colonies to specialize further. The idea is that you grow into needing the services provided by these structures. We already have some of that with the way that the tradehouse works.

On the kingdom level, there could be another structure that provides even more advanced functions required for managing a kingdom, perhaps providing advanced interfaces used for balancing resources and populations across many settlements. Call it a palace and have it make the hosting city your capital, and maybe provide local bonuses as well. You could then take this further to the empire level, perhaps as some major long term upgrade or series of upgrades to your palace, with perhaps further regional administrative structures as well. They could also provide broader ui elements, such as markings on the world/region/empire/whatever map. Or maybe you don't start counting the year number until you build a palace and found a kingdom.