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I've read on Wikipedia and in other sources that the lead position alternates due to flight fatigue, but does it alternate between a small cabal of of leadership birds, or does (almost) every bird get to sometimes go first?

Is the process different across different species?

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    $\begingroup$ Since this question is quite old (it jumped to the first page because someone just answered it) you probably won't read this comment, but it's worth mentioning this: the "lead" bird (at the front) is not the "leader" bird (who decides the direction). Normally, the "leader" bird is somewhere at the middle of the V. $\endgroup$ – user24284 Sep 26 '17 at 0:42
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There is an advantage for birds to fly in a V-Formation as stated by Portugal et al, 2013:

As a bird flaps, a rotating vortex of air rolls off each of its wingtips. These vortices mean that the air immediately behind the bird gets constantly pushed downwards (downwash), and the air behind it and off to the sides gets pushed upwards (upwash). If another bird flies in either of these upwash zones, it gets free lift. It can save energy by mooching off the air flow created by its flock-mate.

There is no advantage to the lead bird though, only the birds behind will get these free lifts by being in the upwash zones.

A study on Ibises determined that there was no clear lead bird and birds cycled between positions. Flying in a V-formation is a collective effort, there would be no benefit to always being in front, so the birds take turns. I would assume the same aerodynamics and the cycling of lead bird apply to all birds that generally fly in V-formations such as Geese. Otherwise the lead bird would just exhaust itself and not have enough energy to complete the migration, it would be the same situation as if the bird was flying alone.

By carefully observing the flock, the investigators learned that the amount of time a bird is leading a formation is strongly correlated with the time it can itself profit from flying behind another bird.

By working together, a flock of birds is greater than the sum of its parts. By not being selfish, individual birds can reap the collective advantage of aerodynamic wash-up, while also allowing for larger flocks, which result in yet more time for an individual to be in a backdraft situation.

Additional Reading:

http://phenomena.nationalgeographic.com/2014/01/15/birds-that-fly-in-a-v-formation-use-an-amazing-trick/

https://io9.gizmodo.com/why-birds-take-turns-at-the-front-when-flying-in-v-form-1683465294

References:

Portugal, Hubel, Fritz, Heese, Trobe, Voelk, Hailes, Wilson & Usherwood. 2013. Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature http://dx.doi.org/10.1038/nature12939
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