Whoever has the initiative ventures into the Undercity when they take it and at each of their upkeeps, until combat damage takes it away.
The rule
The initiative is a designation, and only one player can have it (CR 726.1 to 726.4). Taking the initiative makes you venture into the Undercity, and so does the beginning of your upkeep while you have it. The Undercity is a dungeon (CR 205.3p, 309).
Whenever a creature deals combat damage to the player who has the initiative, that creature's controller takes it, and ventures into the Undercity at once.
If the player with the initiative leaves the game, the active player takes it.
In Life Counter
The initiative is a token on the table. Drag it to whoever takes it. When its holder is out of the game, it goes to the active player. Track the room you are in with the dungeon counter.