If you open the game after the event timer for a Travels Trip has run out and you still have unclaimed rewards, a screen will automatically appear so you can claim those rewards.
If you are on the Travels screen when the event timer runs out, you can still claim unclaimed rewards afterward and even purchase the Premium Pass. However, you won’t be able to start another story sequence. Once you leave the Travels screen to return to the island, the Travels button will no longer show up on the island screen.
If the event timer runs out while you are playing a story sequence in Travels, you will still be able to finish that story sequence and claim your unlocked reward afterward.