There is fast track, which is more positive than neutral.
In your first example, you could say:
The student was fast-tracked from grade 3 to grade 6.
fast track (verb, with object)
accelerate the progress of (a person or project)
This is often taken as evidence that gender discrimination persists, and as basis for calls to fast-track women to higher positions.
She was lovely too, said she'd fast-track us through the selection process to end up ‘on-air’.
At the hospital, which is full of people who look like they have been waiting around for an awfully long time, I am efficiently fast-tracked
through the system.
Despite his protestations of innocence he was fast-tracked into court the following day and jailed for 11 years.
(In the example sentences above, I have only included those that relate to people rather than projects.)
Alternatively there is pluck from the ranks, which is more neutral as to whether or not this promotion was a good thing or not, and perhaps implies more sudden, rapid and arbitrary a rise, and less of an ongoing or regular process than with fast track.
In your second example, you could say:
The clerk was plucked from the ranks to fill the general manager position.
It is obviously a military analogy. I can't find a specific dictionary listing for this phrase, but it is the second-listed sense of pluck in this definition:
quickly or suddenly remove someone from a dangerous or unpleasant situation
The baby was plucked from a grim orphanage.
There are other promotion-related phrases based on from the ranks, such as:
all of which imply a slow process, driven by one's own efforts - definitely not what you require! However these phrases provide the context for pluck from the ranks.
Examples sentences for the phrase pluck from the ranks can be found easily by a Google Books search. To take the first three examples from this search with complete sentences available:
Four & Twenty Blackbirds, Mercedes Lackey (1997)
"I could read those and give you a summary," said her secretary Kayne Davenkent, a clever and steady young novice that Ardis had plucked from the ranks of the scribes just last summer.
The Far Side of Eden: New Money, Old Land, and the Battle for Napa Valley, James Conaway (2015)
She saw that they would attempt to knock her off, just as they and men like them had knocked off women supervisors in the past, with money and an inexperienced, pliant good-old-boy candidate plucked from the ranks of commerce.
Meet the Next President, Bill Sammon (2007)
Thus, it is entirely possible that the next vice president will be plucked from the ranks of presidential candidates who fall short of the top prize in 2008. Just don't expect any of these candidates to actually admit their strategy in advance.