Timestamps:
2379 (TNG, some point before Nemesis)
2377 (VOY, "two years earlier", still in Delta Quadrant)
2371 (DS9, "eight years earlier", between The Adversary and The Way of the Warrior as Sisko was just promoted to Captain but already shaved his head)
2269 (TOS, "110 years earlier", possibly before TAS as Arex or M'Ress aren't seen, but neither is Chekov, possibly)
Some memory loss or time travel must be involved, as Sisko only encountered Q once (Q-less), according to Picard, and Kirk of course didn't know who Picard was in Generations (unless they are protecting the timeline).
Crew lists, for the Conflict:
TOS: Kirk, Spock, McCoy, Scott, Sulu, Uhura, nondescript yellowshirt (Chekov? Our only casualty?)
TNG: Picard, Riker, Data, La Forge, Troi
DS9: Sisko, Kira, Jadzia, O'Brien, Odo, Quark
VOY: Janeway, Chakotay, Tuvok, Paris, Torres, Kim, Seven
That's 7-5-6-7, so to balance their may be some unspotted characters there.
The four pick Captains (Trelane- Kirk, Metron- Janeway, Ayelborne- Sisko, Q- Picard), but the sponsors will select the rest of the team outside of crew affiliation. So the 25 known could be split into teams of 7-6-6-6.
Shame they couldn't get Enterprise involved. The Organians could've picked Archer instead, with I guess Thasians or something choosing Sisko.