What is the ratio of Tourism to Culture when influencing a civilization?

Culture and Tourism match up 1:1, and their numbers are considered over all time; that is, if your opponent has earned, say, ten thousand total points of culture since the start of the game, you need to have ten thousand points of tourism against them to become dominant. If the tourism/culture indicates that your influence over another civ is rising, then you are generating more tourism against them than they are generating culture.

Note, however, that even when you are only showing four points of tourism in the bar at the top, you may be generating more against specific civilizations; open borders, trade routes, religion, social policies and more can give you tourism bonuses against specific civilizations that are not reflected up there.

You can get a pretty good feel for all this by opening the Culture Overview. Just click the Tourism indicator on the top bar, and you should see a window like this (you may need to flip over to the Influence by Player tab):

In this window, you can see, for each civilization, what level of influence you have, what your Tourism multiplier is against them, how much total net tourism that's resulting in per turn, and how it's trending. More interestingly, you can hover over the Tourism and Culture bars in the middle section to see the current net totals of each, and also if the trend is "Rising", you can hover over that arrow to see an estimate of how long it will take to reach the "Influential" level, assuming your tourism rate and their culture rate remain constant.

Furthermore, even if you're not making a play at a Culture Victory yourself, this information is still useful; hit the Select Player dropdown, switch to another player, and you can see how well their tourism is doing, both against you, and against other civilizations.

The phenomenon you described above is simply a mathematical issue. I believe that the "rising" refers to the percentage (of your total tourism compared to total culture, not whether you are producing more tourism then your enemy produces culture per round.

After all, you normally shouldn´t be able to outproduce other civ´s culture with your tourism until quite lategame, where powerful bonuses kick in (Hotels, Airports, National Visitor Center, Internet), and especially not your example of 4 Tourism, I can´t believe an enemy would produce only 3 Culture in what probably has to be at least round 75 on Standard?

But the "rising" tab referring to Influence percentage (aka 100*your total Tourism/total other civ´s culture) makes sense to me: Considering that the influence states (Unknown, Exotic, etc.) are measured as percentages, it makes sense that you want to know whether your and other influences are rising or falling (aka do I have to fear more ideological pressure or can I reduce it soon?)

This also explains why your influence could "rise" with just 4 Tourism, despite that your enemies should all be outproducing you with something like 15-50 Culture. So why is your percentage rising, although the per turn value is lower, which means that the amount of Tourism you are behind other civ´s culture is increasing (aka actually putting you further away from a cultural victory)?

That´s because the influence percentage (your total Tourism vs others total Culture) will adjust a little bit to the current percentage of your Tourism/Turn divided by enemy Culture/Turn. Example: You produce 10 Tourism/Turn, another civ you have just met produces Culture/Turn, but has already produced 10000 total Culture. Assuming the per Turn values don´t change, each Turn will put you a further difference of 90 (in addition to the starting 10000) away from becoming Influential (100%). You start out with 0% Influence. But for example 900 Turns later, you will have produced 9000 Tourism, they have additional 90000 Culture, so a total of 100000, which means you have 9% total Tourism of their total Culture. So your influence percentage was rising all the time, but now you are not only 10000, but 91000 points away from getting Influential, and until you manage to raise the per turn percentage of yT/oC, you will never be able to advance past 10% influence, although your percentage will rise minimally.

This would explain, why most of the time your influence percentages might be rising, since generally, the per Turn percentages will increase, remaining at 0% until you start to generate Tourism, but hopefully, otherwise you will never win a cultural victory, at some point you manage to get more Tourism than the enemies get Culture per Turn, and only then will you actually be able to reach those 100% influence (well, it still is a matter of waiting, increasing Tourism some more and surviving) and win.