My experience has shown me it has do with the circumstances of that particular elder.
Area 1 - Removing
Elders that are removed are usually for the heavier matters.
1) I have heard/seen elders removed where they are never to be re-appointed, eg causing divisions in the congregation, apostasy etc.
2) Elders can be removed for other serious judicial reasons, eg fornication etc. (Sometimes the body may recommend the elder to voluntary stepdown if he happens to confesses the sin before it becomes public knowledge, thus saves them the embarressment of being removed.) Depending on crime depends on how they can be re-appointed.
Majority of cases there are usually appointed as MS, then at a later date as an elder. 2-3 years time period is usually the standard period, but things can and do happen quicker for many reasons. At times, they may be reappointed directly as an elder without being firstly an MS. Rare, but there have been cases have happened in the past.
Area 2 - Stepping Down (Voluntarily)
1) An elder can step down for few reasons, eg family commitments, sickness etc.
2) Some find that being an elder is not what it cracks up to be, or is out of their capabilities (eg compassionate, long-suffering with the brothers, etc).
Most common for a elder to step down is usually sickness or family commitments.
These either have the option where they can go back to being a MS if they feel that they can still handle to 'running of the business' responsilities, but they dont want the teaching/shepherding responsilities. Alot of environmental factors can play in this area too, eg not enough MS in the cong, or they have plenty but they do not wish the elder to feel lonely etc.
Most of elders that step down, just wishes to be a publisher until things are back in order. When the time is right again, they can be either be appointed as an elder or MS.
Area 3 - Stepping Down (Forcibly)
1) An elder that has rocked the bandwagon can be forced to step down by others (especially those who want a tight grip on the cong).
2) Some elders step down cause they cannot get along with the "Body" and find it better to stay out of the circle.
In most of these cases, once the elder steps down, they usually will not reach out for any priviledges ever again. True the saying, "once bitten, twice shy".
Others that want to reach out again either:
a) are just wanting to get back at the body
b) are just plain suckers for punishment
c) are genuinely true elder material, but wishes to get on with things and makes it a learning lesson. (Very, very, very rare)