What I don't get is how you're supposed to win culturally against a 'wide' player that builds a few culture buildings in each city; you can easily skyrocket above 100 culture per turn - how in the world do you catch up to that?
100 CpT is really not much end game. Also remember that tourism gets additional modifiers to your "base" on a per civ basis like 25% for open boarders for example. By end game with internet 600 or 700 tourism isn't hard to reach and when you add modifiers, 1000 on civs is easily feasible. Unless they are doing 1000 CpT, you will eventually catch up to them.
Regarding the rising/falling. Its completely based on the % influence change. 1 tourism per turn compared to someone gaining 50 culture per turn will still show rising on the graph if you are just starting out at 0 tourism. Basically your influence can reach 2% at 1 tourism to 50 culture per turn so until it gets to that point, it will show rising.
About the only times it will ever show falling is if someone just burned a writer for culture. Since the large change lowered your influence %, it will show falling for that one turn. If its over multiple turns it means they are prob under a golden age and thus the % is going down slightly.
Like for example.
You generate 50 tourism and have generated 6000 tourism so far. Against someone generating 150culture per turn and has a total culture of 24000. Assume no modifiers to tourism against that civ. Right now you stand at an influence level of 25% (6000/24000). Your current theoretical max influence is 33.3% (50/150).
Next Turn assuming values per turn stay the same, its 6050 tourism to 24150 culture. Your influence has risen to 25.05%. 9 More turns pass with no change and its now 6500 to 25500 (influence has risen to 25.49%).
Now the next turn the culture civ entered a golden age and had a few culture buildings finish. Their culture per turn rose to 200. So on that turn, the levels would be 6550 to 25700. Your influence actually fell to 25.48%. Thats because your theoritical max has fallen from 33.3 (50/150) down to 25% (50/200) so your influence will fall towards that point until you either increase your tourism or in the above case, their golden age ends.
The changes are more pronouced when you factor in the modifiers as say you had open boarders and the deal ended, you just lost 25% tourism being generated to that civ so their CPT could outstrip your modified tourism influence.