Presidents are selected from state governors. Governors are selected from mayors and city managers of large cities. They then from small cities.
Similarly several levels of legislatures and legislators.
Similarly several levels of judges.
All this is already done mostly informally. New wrinkle: each level periodically shuffles its members, e.g., governors move to governing a different state. This demonstrates how they perform under broadly varied conditions, useful information when selecting for the next higher level up the hierarchy.
Flaws in the system:
Only the president ever deals with military, monetary policy, international diplomacy.
Local people might not be lead or represented by one of their own, if things keep getting shuffled. Maybe apply only to the executive and judicial branches.
Is corruption worse under this system?
By requiring candidates for a higher post all to have experience at a lower post, it allows comparing ability: who did better governing (say) a poor state? Judging by ability is fine, but then where and how does politics -- differences in opinion -- enter? (It will inevitably do so. Perhaps corruption.) It would be nice if the legislature could be kept the only political branch of government, but seems unlikely.
Political enemies might try to make go poorly governor's term (hurting the residents in the process) for the larger prize of derailing that person's chances at presidency.
Being effective at a certain level of the hierarchy might require a lifetime to learn. There aren't enough lifetimes to learn the next level.
What are the required qualifications for Vice President? Maybe Lt. Governor. Then, we assume vice-presidency qualifies you for presidency.