These projects employed skilled journey men (level between apprentice and master). At some point they then traveled to work on the next project, spreading the know-how. Once something was working, it was often copied as modifications were risky.

I'd rather call the knowledgeable entity in question "expert" than company, but I was wondering along similar lines: would they travel far to projects or would the knowledge be dispersed to enough people that most projects could just happen with local expertise? (plus temporarily imported capacity for big project, but not necessarily imported expertise?)

What's clear is that knowledge transfer (from region to region and from generation to generation) was facilitated by traveling apprentices, a tradition that still lives on and (I believe) is documented to go back to that time. Their profession wouldn't be bridge builder but merely some contributing role (with the occasional exception of whatever the period-correct term for architect or project manager would be), but of all the masons involved, you'd have some whose learning circuit involved a bridge, of the woodworkers you'd have some who dabbled in cranes before and so on. But was that usually sufficient or was there also a pattern of on-demand traveling leadership?

edit: post was lingering in the input form for some hours, some good answers already there!