I have talked with a friend that was the best in class PhD student at a top-200 university in USA, but she couldn't find any position as assistant professor position or fellowship in a top-20 university.
I'll assume you are talking about tenure-track positions, due to the reference to an assistant professorship. If you are talking about postdocs, then it really depends on how your field works (e.g., how many such positions there are and how they are distributed among different types of universities).
It's certainly possible to get a job in a much higher-ranked department than the one you graduated from, but a little arithmetic shows that it's very unlikely, assuming departments in your field aren't growing dramatically.
Let's assume a steady state, in which departments are neither growing nor shrinking, and let's make the simplifying assumption that they are all the same size. Of course these assumptions aren't quite right, but they can generally serve as a first approximation.
What does it take to get a job at a comparable department to the one you graduated from? In a steady state, each professor will eventually be replaced by one new hire, but they will typically train multiple Ph.D. students. This means that there's just enough room to hire each professor's best student ever, over their whole career. Of course the hiring won't actually work out that way, since the top students are not uniformly distributed among professors (and there will also be competition from students who attended higher or lower-ranked schools). However, this sets a lower bound: if you aren't at least as good as the career-best student of a typical professor from a top X department, then you can't realistically expect to get a job at a top X department yourself.
Going up in the rankings is even harder. The number of people who are their advisor's career-best student at a top 200 department is ten times the total number of positions at top 20 departments. This means that 90% of them will fail to get such a job, since there just aren't enough positions to accommodate them. In practice, it's even worse than these numbers indicate (generally much worse), but the point is that simple arithmetic guarantees at least a 90% failure rate even for these highly successful students.
So you shouldn't be surprised that your friend failed to go up in the rankings by a factor of 10: that requires either exceptional luck or exceptional accomplishments.
What is taken under consideration when one makes such application?
It's really no different from any other application, with no special requirements or considerations. You need to make the case that you are a better choice than any other available candidate. Applicants from low-ranked department often face some initial skepticism, but hardly anyone would be silly enough to obsess over rankings while disregarding visibly superior accomplishments.
Graduating a PhD program is completely different from previous levels. Potential employers in academia are looking for someone who would add prestige to their university. If this is a research university, this means someone with lots of highly ranked publications and a history of bringing in grant money. After that, you need a good professional reputation (and the strong recommendation letters that come with it), and an affable personality.
GPA or individual ranking within the PhD program does not matter. At all.
This is based on my experience on a hiring committee for an Ivy League institution in the USA. I would add that at this particular institution, they would only entertain applicants who had earned PhDs from the top 10 or so universities in the field. Overall university rankings were not important; only within the same field.
What is taken into consideration: the PhD thesis.
Unless the thesis is a "top 30" caliber work (a little leeway), you probably won't get a "top 20" appointment. Many of the PhD graduates from "top 20" departments will also end up a lower ranked places. Those top departments produce far more PhDs than they hire!
Rank 200 to rank 20? It has happened. But rarely.
Echoing some points from other answers, and from my comment to the original question:
In my experience, in mathematics at an R1 place in the U.S., people on postdoc or hiring committees do not think in terms of "rankings", either of departments or of students within their graduating class. Nor grade-point average.
Rather, to some degree, they do look at the substance of a candidate's work, although probably stopping short of reading the thesis or papers. Rather, most often, they/we look at the letter writers' appraisals. These are most persuasive if the letter writers are personally known to the committee, or are well-known in the mathematics research milieu, and are expert in the subject at hand.
However, in any case, there are far more PhD's being produced by "the top 200" programs than there are positions available in the "top 20". As already observed, for similar raw numerical reasons, most graduates of the "top 20" do not get "top 20" jobs.
In the past (e.g., pre-internet), there was also a common disadvantage that the level of awareness at "top 200" places might have been very different from that at "top 20", affecting the nature of the projects attempted or completed. It appears that this effect is diminishing, also due to the eflux of new PhD's from the most active research centers to nearly every branch university branch campus in the U.S.
You need to at least have a postdoc from a University of the same rank. I'm not sure where the ranking of top 200 came from, are there even 200 PhD granting institutions?
