Having spent time watching some of today's leaders who are requested to write haskamos I can tell you the process I have seen.
Situation A. A person the Rabbi/Rosh Yeshiva knows closely, either from the yeshiva or the neighborhood brings in a manuscript and request an approbation. The Rabbi/Rosh Yeshiva happily obliges, sometimes while flipping through the papers, casually asking what it is about.
Situation B. A person the Rabbi/Rosh Yeshiva never met comes in and introduces himself as Rabbi Nick Nud and requests an approbation. The Rabbi/Rosh Yeshiva tries to excuse himself by saying he doesn't give approbations, all without asking any questions about the work or looking at it. From here there are multiple outcomes. Either Rabbi Nud is smart enough to get the hint, he leaves, sometimes to return as will be mentioned later in situation C. If Rabbi Nud doesn't get the hint, he will spend the better part of a day chasing and hounding the Rabbi/Rosh Yeshiva who almost certainly will win this standoff.
Situation C. A person walks in and shows a list of approbations he has been accumulating, starting with his local friend Rabbi and continuing up the ladder showing each successive Rabbi the previous one's letter basically saying well he signed, so you should too. This procedure seems to work pretty well. In fact, sometimes Rabbi Nud, our friend from situation B, retuns after having gone through these steps and actually will be successful in securing his top tier dream Rabbi approbation. Yay!