Perhaps Indian author Vishal Mangalwadi's example of the mathematical theory behind the mechanical clock would illustrate the flow of ideas well. He wrote (I'm going my memory here, so I don't have his exact dates) that an Indian mathematician came up with the theory for a mechanical clock in the early 11th century, but didn't try to build one; 50 years later middle eastern Muslim scholars were debating and studying the theory, but didn't try to build one; another half-century later the idea had come to Europe, and it was there in the early 12th century that the bishop of Paris suggested to his monks that building a mechanical clock would improved their ability to organized their communal work and worship in their monastic communities.
Mamgalwadi also asks the question of why neither the Indian nor Islamic cultures tried to build a mechanical clock when they knew of the theory. He suggests that the Indian Hindu belief that reality is /maya/ or illusion (and one meditates to escape the illusion) prevented them from trying to make a clock; in the middle-east, he suggests that the Islamic rejection of possibility of God becoming incarnate in the person of Jesus prevented them from trying to flesh out the theory, while in Christian medieval Europe their belief in incarnation primed the culture for trying to work the theory out in practice.
I think you've missed Al-Jazari who's well regarded as the father of modern robotics several hundred years before Leonardo da Vinci [1]. Some historians even suggesting that some of the da-Vinci inventions were copycats and derivatives of the Al-Jazari's more than hundred of inventions but he's not properly credited by da Vinci [2].
FYI, of al-jazri most famous invention is the elephant clock [3].
Mamgalwadi also asks the question of why neither the Indian nor Islamic cultures tried to build a mechanical clock when they knew of the theory. He suggests that the Indian Hindu belief that reality is /maya/ or illusion (and one meditates to escape the illusion) prevented them from trying to make a clock; in the middle-east, he suggests that the Islamic rejection of possibility of God becoming incarnate in the person of Jesus prevented them from trying to flesh out the theory, while in Christian medieval Europe their belief in incarnation primed the culture for trying to work the theory out in practice.