It looks like the rule book specifically says turn order is tied only to the movement action space and the dice stacked on the the board. There are a few spots where it's vague though and seems like any other type of movement may also move you up in position.
What is the rule?
Pretty sure it is just the paid movement space (two dice, coins based around distance traveled). That's how it works in Yucata as well.
If it were other forms of movement it would be pretty imbalanced if the city actions only available to some could count. Imagine one of those is in Sumantra and someone starts in Beijing. Until someone else got there they could lie in wait and always claim start player.