I think Influence, as far as borders, should be more of a "zone of control" type of thing, which can be spread by building new forts, settling a small village, and having patrols or border garrisons. Im not entirely sure it should be a cultural thing, like in Civ 4, as typical culture buildings like theatres, ect, would seem more likely to gain prestige.
Although, I could see certain things increasing both prestige and music. Say, an established musical tradition could increase influence ... if the peoples in the countryside are willing to take up arms to defend the musical tradition. While, say, a grande bard was to hold a grand performance at the Theatrical Opera, it could be either a temporary or permanent addition to the prestige of the building.
I do agree that the most significant population factor in the short-term is Population movement by prestige, but how hard would it be to have a minimal natural child-birth growth factor, and although its minimal, it can still add up to something significant as cities are getting larger. I mean ... I think its kind of de-facto that most places are having positive growth via child-birthing, but I think while early on the majority of child-births would be present in the wilderness, as more and more people enter civilized culture, and there is less uncivilized wild-men, the majority of births will take place in large city centers and in civilized rural areas.
Wild-men, however, will naturally have much shorter lives, and increased fertility, merely as a genetic survival mechanism. Im not saying their genetics are different, but when living out in the harsh out-lands, your population will most likely experience more deaths, and as a natural defense mechanism the same population will have a much higher fertility rate than those that have settled down for jobs. Not to mention the fact that those settling in cities are living more for wealth, prestige, and pleasure, and making babies tends to limit one's ability to pursue such things. Which is why various forms of attempted abortion have existed in structured civilizations since the invention of agriculture. Abortions and Abstinence, combined with the genetics feeling safe in a structured environment, lead to a limited growth-rate via natural births in civilized cultures. As a rule the more "intelligent" and forward-thinking a people, the more obsessed they are about delaying the natural process of child-birth.
Even with higher fertility rates in the wild, however, once there is an abundance of food and health in cities, There will be more child-births in the Cities than there will be in the wild. I think a sophisticated, perhaps geometric, equation for child-birth will be suitable.
I will attempt at giving a crude example: Natural birth rate in a city of 1% for every 10 turns. For every 100 kilograms of spare food able to be produced by the farm-land surrounding a city, base birth-rate is increased by 1%. If you want to increase population each turn, chance the birth rate to 0.1% each turn for every spare 100 kilograms plus a 0.1% base growth-rate. Another genetic mechanism responds to recent deaths. If more than 10% of the population has died, then the birthrate is increased by 1% (on a turn basis). So from a turn basis, 0.1% for every extra bushel of food, 1% if more than 10% of the population has died (in the past 10 turns) ... so that would be 1% extra for 10 consecutive turns. In addition a nursery would increase birth-rate by 1%, and a brothel would increase birth rate by 0.1%. Some sort of civic which encourages parenthood and births would increase the birth-rate by an additional 1%. I just see this as an underlying Organic mechanic to make extreme situations flow more naturally. (increases in birth will be rounded down to the nearest whole number)
Obviously there are many people that can come up with a better form of natural birth-rate than I just laid out. However the system works, it should not really allow for any more than 1 super-city per nation, and each sucessive city should some-how be about 30-70% the size of the previous city. Not sure how to do that, but I don't want all size 10 cities or what-not. I want one size 20, one size 12, one size 8, one size 5 one size 3 and alot of tiny size 1s