The travel referred to is presumably from the story about the census of "all the world" in gLuke. There's no strong reason to think that this journey happened at all, and good reasons to think that it didn't. First, the census that motivates the journey makes no sense. read more
The Protoevangelium of James, an apocryphal Gospel probably written about AD 145, Says that Joseph used a donkey to bring Mary to Bethlehem The day of the Lord shall itself bring it to pass as the Lord will. read more
Mary Riding the Donkey The Bible gives very little detail on Mary & Joseph's actual trip to Bethlehem. Considering the times and the location, though, it was very likely that they would've had a donkey carrying their belongings on the trip, and Mary, who was well along in her pregnancy, would've ridden it rather than walked the entire way. read more