Kinda complicated, but let me try to explain what I've learned from playing.
For s2, you need approximately 4 kills per player. However, when a play joins or leaves, the effect is not instant. When a player joins a team, 1 kill is added to the counter immediately, and it slowly rises to that 4k per player. When a player leaves a team, there is no immediate effect, but it will slowly fall to that 4k per player. Stage 3 works in a similar fashion, but will require twice as many kills.
When the number of stagekills reach the number of kills per stage, a stage will advance. Server ops with rcon can adjust each team's stage-kills with g_[team]stagekills. If a team has their stage adjusted downwards by server op, they will display "0 kills to next stage" and will earn the stage with their next kill.
Server operators can change the kills needed with g_[team]stage
- kills, but as the formula is somewhat unknown, these variables are not often changed.
Got all that?