The title says it all.
How do the other Wizarding schools find their witches and wizards?
Do they have a different version of the Book of Admittance (and Quill of Acceptance), and is there a boundary between the schools where each book "stops"?
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They have various means, but since we've only seen the UK view thus far, we only have firsthand knowledge of the Book of Admittance from the UK & Ireland. Nothing has been revealed about how the other schools identify their prospective students. However, it is known that most wizards homeschool, so our view of Wizard schooling is actually NOT the norm.
We also know that Beaxbatons traditionally accepts students from Western Europe, and Durmstrang from Eastern Europe and the Scandinavian Peninsula... this could indicate many things for identification of students, including a location-based identification (leylines, et cetera), or even language-based (That is, Latin-based vs Non-Latin-based).
Uagadou is the largest of several wizarding schools in Africa, and they reach out to prospective students via Dream Messenger. It is an extrapolation, but not a large one, that student identification is based on some sort of shamanistic practice.