In this case, to emphasize that the way of thinking of each individual person in turn will be examined, you can use:
Encountering a succession of persons to observe how each one's way of thinking affects:
- their way to look at the world
(emphasis added). "Their" is (here) a perfectly valid singular pronoun to use when you don't know (or don't care to specify) whether someone is male or female1, 2. "His" also works, or worked, for that, but it's falling out of fashion* and in this particular case it is more awkward than usual because of the context being an entire collection of people.
Alternatively, you can replace the bolded word, which is rather over-formal, with "people", and that should also work out okay; even though it isn't quite as individual-focused, the context still makes it clear enough, and it's a lot smoother in most styles.
"Each one's way of thinking" is a little awkward, but there's nothing really wrong with it and no obvious substitute. Don't worry about it.
*In part because it sounds sexist and in part just because it sounds odd.
1 OED Online, sense 1.1
2 Merriam-Webster, sense 2