After the match
Rematch, who is host and how a party's warm-up differs from quick play.
Rematch
When a match ends, the host sees a Rematch button. Everyone else sees "Only the party host can start a rematch". A rematch restarts in the same room with the scores reset, skips the warm-up and replays the maps in the same order.
Quick play has no Rematch. There you get "Find New Match" and Leave.
When the host leaves
If the host leaves, the player who has been in the room longest takes over as host, with the host's powers. A short disconnect doesn't end the party: the room, its bots and the host's seat are held for a moment so you can come back.
The warm-up
A party's match begins with a warm-up on its first map: fists only, until the first kill that counts. A party has no time limit on it, so it ends when someone gets that kill.
Quick play is different. A quick-play room with one human and bots starts the match by itself after 10 seconds.