Ahh, now that I re-read Warriors of Kerensky (p45-46), it explains things more clearly. I don't know how I missed the last paragraph in the section.
Every warrior undergoes at least one Trial of Position when he passes from cadet to full-fledged warrior, traditionally on or around the cadet's twentieth birthday. Several Clans graduate warriros at eighteen, however, and extreme circumstances may prompt Trials of Position even earlier.
That's pretty crystal clear, although we still don't know
which clans graduate people at earlier ages. The paragraph goes on to clarify which clans allow a re-test and what happens if you fail.
Warriors are required to retest frequently throughout their careers (...) However, role and assignment offer a degree of flexibillity, and combat performance is frequently accepted in lieu of a formal Trial. (...) Those who perform well "test up" and are promoted. Those who do not either retain their current status or "test down" and are relegated to second-line units and non-combat posts or demoted to a lower caste.
These annual retrials, which unlike the first, need not involve 'Mechs or battle armor, or often more akin to Trials of Possession, with senior officers fending off challenges by their juniors and peers competing for limited promotion slots.
There's a lot more, including that you need the Clan Council or Khan to approve these challenges.
This is
very vague, but I think that's because different clans do things differently. In many clans, you can challenge your way up the ladder virtually unhindered. The top officers aren't necessarily the best strategists or administrators, but they are the best combatants. In other clans, like I'd imagine the Star Adders, Diamond Sharks, and Wolves, the Khan and Clan Council restrict access to trials to protect valuable senior officers, and can use the approve/disapprove rule to keep freebirths,
dezgra warriors, and politically unpopular leaders out of top positions even if they rock in the cockpit.
The history of the Clans is filled with examples of great warriors who run their clan into the ground because nobody can beat them in a challenge to replace them. My earlier speculations notwithstanding, Khan Ian Hawker of the Diamond Sharks is one obvious example-- after a while the Sharks were desperate to replace him but couldn't due to his skill as a MechWarrior. Similarly, the crusaders Diamond Sharks hammered saKhan Angus Labov to unseat him from Gamma Galaxy... and this illustrates another aspect of the process: the losers were getting relegated to garrison units as punishment for losing. He ultimately stopped accepting challenges.
So I think it's probably similar across the clans in its broad outlines but how it happens varies from clan to clan and situation to situation.