The party moves from day to night, each time they move forward to a new day they get time tokens. Time tokens represent lead time on whatever pressing task needs to be done.
When you run out of time tokens, you are about to get caught, or miss the wedding or whatever. So we start rolling a percentile die to see if "the bad thing happens"
Every morning you make an exhaustion check:
DC 18
-2 if you ate
-3 if you were warm
-3 if you had water
At disadvantage if you didn't sleep.
You can spend time points to sleep, or search. When you search, pick an Event DC and the DM picks it up to figure out what kind of check it requires. Each character that searches must spend 1 time to do so.
(Most checks are survival or have a few options/approaches)
Events can be good or bad. To search and find a good event usually means finding resources or shelter. To search and find a bad event usually means you spot it before something bad happens to you.
When the party token moves to the next time, the DM will check to see all of the events to see if anything bad or good happens without the players noticing.
If the characters are interrupted during their "sleep" phase, they can choose to "get up and help" or keep sleeping. If they get up, they lose the benefits of a long rest.
This can be a very interesting way to deal with a journey, especially if it's on time pressure, and has a pretty set path. If you wanted to give players different choices, you could also conceivably make different event setups depending on their path of choice. If there isn't time pressure, the system tends to fall apart a bit. The time tokens only work if there is a "risk-reward" element to spending them.