One important lesson I've learned about multi-institutional collaborations involves alignment. It's critical from the beginning to not be in a rush. A collaboration needs to have a clearly aligned purpose behind it.
Part of the issue here, I think, is that people confuse the word collaboration with the word cooperation. In common language, we use those terms almost interchangeably, but they're very different concepts.
Collaborations are hard. They require a lot of energy. They bring huge benefits, but no one should enter into a collaboration lightly. You can enter into cooperation lightly, because it's always good to be cooperative, but you need to be much more intentional about collaborations.
A second lesson I've learned about multi-institutional collaborations involves how challenging it is to design collaborations and systems that are adaptive, that respond to the fact that the world is fundamentally a complex system. Whatever you plan, there's going to be a change somewhere down the line, and you're going to have to respond to that change.