Dangers and threats, duty and responsibility, increased pressure and ticking clocks

Crossing a rope bridge that's fallen into disrepair
Traversing unstable ground (due to a mudslide, loose rocks, a steep grade, etc.) 
Wading or swimming across a river with a strong current
Driving over a partly flooded road
Navigating unseen hazards (a minefield, a snow-covered crevasse, pockets of radiation, etc.)
Sneaking across a guarded or patrolled area 
Traveling through a predator's territory  
Enduring rough seas and storms on an ocean voyage
Crossing a desert or other desolate location where resources are sparse 
Moving through a booby-trapped room
Walking a precarious path in the dark
Escaping an oppressive and dangerous region where leaving is not allowed

Having to travel light and leave behind anything unessential 
Poor planning resulting in the trip taking longer than anticipated
Needing to move quickly; having to push on when rest is needed
Leaving behind loved ones so they'll be safe 
Being on edge, alert for danger
Suffering scrapes and bruises from having to move quickly
People questioning the character's leadership choices
Needing to blend in and having difficulty doing so
Navigational challenges (traveling at night, driving without headlights, etc.)
Having to wait until it's safe (for a guard to move on, a hazard to be dismantled, etc.)
Losing something during the crossing and being unable to go back for it
Traveling with companions that make it harder to move undetected (babies, sick people, etc.)
Picking up more people along the way
Members of the character's party not getting along