I don't see why having stuff happen when travelling needs to be significantly different to having stuff happen in a dungeon or city.
In addition, you can have 'resource management' problems when travelling, such as running out of food and having to hunt. The problem with this is that D&D has always made 'miscellaneous equipment' such as rations so cheap that the players should always have enough. If players had to choose between buying armour and buying a horse, and the lack of a horse was as potentially dangerous as the lack of a sword, then that might make for more interesting choices.
The board game 'Oregon Trail' is set in the old West, but it might be a good source of ideas. Message me if you want a copy of the rules.