While doing some research for another question I found out that there is a spell that can propel boats forward:
"Seems a shame ter row, though," said Hagrid, giving Harry another of his sideways looks. "If I was ter — er — speed things up a bit, would yeh mind not mentionin' it at Hogwarts?"
"Of course not," said Harry, eager to see more magic. Hagrid pulled out the pink umbrella again, tapped it twice on the side of the boat, and they sped off toward land.
Harry Potter and the Philosopher's Stone: Chapter 6 - "The Journey from Platform Nine and Three-Quarters"
If there's a spell to propel a boat forward it seems incredibly likely that there is also a spell that could move a carriage.
So why does Hogwarts use Thestrals to pull the carriages and not magic?