The bowed instruments traveled along the Silk Road. There's no doubt about that. I think the earliest stringed instruments such as the African ones didn't require a lot of woodworking just a bow and a resonance chamber like a gourd or a tortoise shell.

What evidently happened is that the nomads came into constant contact with agrarian settlements and trade-offs occurred. I think they became similar to Vikings--having farms of their own but also going off on long journeys to trade and raid. But the Silk Road would have brought them in contact with all kinds of cultures with materials they came to need more and more eventually had to learn to produce it themselves.

We see the same thing in North America--the Navajo learned to raise livestock, the Hopi were farmers but the the Apache (who are closely related to the Navajo) were nomadic raiders (their name means "enemy") and very skilled horsemen.