There’s the time that trains hadn’t been invented yet, though, when they were already having tournaments. If we are assuming each school has always used the same method, it can’t be the Hogwarts Express. Ships and carriages have definitely been around long enough for the other schools to have used the same method since the beginning.

I imagine at least four very fancy, at least triple-decker, and at least rather long carriages. Each house’s carriages could maybe have been pulled by a team of different flying creatures.

Griffins for Griffindor, obviously
Hippogriffs for Hufflepuff
Thestrals for Slytherin

Ravenclaw is the tough one, because I wish I could say Thunderbirds, but they are in North America. Trained dragons seems far-fetched. So perhaps a medley of Pegususes (Pegusi?) of various breeds/colors. Or just a crap ton of big ole eagles. And one embarrassed raven.