The Big Walk save file is created and kept by the host, and world progress saves automatically. To continue the same journey, the original host must start that saved game before the other players reconnect.
How autosave and hosting work
The official FAQ says players do not need to trigger a manual save. When the host starts a session, other players can connect; when the host ends it, all guests are disconnected.
The host selects the saved world again from the host-game flow when the group returns. The host can also choose the two-player, three-player, or four-plus world version for the new session, because that choice is made each time the world starts.
Rejoin the saved world correctly
Same-platform friends can use the friends-game menu after the host is online. Other players use the newly generated Join Code, which changes every time the host starts a session.
Players who join later arrive through the opening area and can reconnect with the group from there. A published save walkthrough also documents a Lost & Found platform near the starting area where carried items from the previous session are returned.
What is not officially documented
The official FAQ does not describe transferring world ownership from one host to another. It also does not publish desktop file-system paths, a manual backup procedure, or a cross-platform save conversion method.
Do not move or replace local files based on an unverified path. The confirmed recovery model is to use the original host's in-game saved world and let the other players rejoin through the current session.