How did a college grant work in 18th/19th century Europe ?

Hello, I am making research for a book, and I need details about grants/scholarships in Western Europe around the 19th century, ideally for a teenager living far away from the applied school.

How would the application work ? On what criteria the grant would be given ? How would the scholar go to the school without the funds to make the trip ? Would someone come and get him, someone appointed to "collect" scholars, or would the scholarship funds be made available to the student so he can go by himself ?

I read some fiction books where the protagonist is a scholar during that time period, but most of them don't mention the scholarship process at all. At best, they remain very evasive on the subject: usually, a private teacher writes a letter at the end of a very basic education (when the student comes to a certain age, or the teacher feels he does not have anything left to teach him), asking for a college grant somewhere in a big city, describing what the student has learned and recommending him. The positive response comes months later, and the protagonist travels to the school. But I fear the books I read oversimplify for the sake of the reader.

I know each country had its own education system and access to scholarship, but I am having trouble finding even very general details. And even though I am targeting Western Europe (especially the scholarships created by Napoleon in post-revolution France), I really don't need a very specific answer: any real details about a scholar travelling to a big city for higher education during that time period would help a lot.

English is not my first language - sorry if I made some mistakes. Thank you for reading my long and very specific question.