Wow, that is one of my own favorite topics.

I would start out with the Chapter "Bow Design and Performance" by Tim Baker in the Traditional Bowyer's Bible, Vol. 1. That chapter will give you a very good overview of the physics involved. An important take away from that chapter is that yew as a bow wood was not a magical material and it was not a natural laminated bow wood. That is a ridiculous myth that is still being popularized by people who should know better (even Robert Hardy).

That said, it is not historically oriented. It is however filled with the most data on very many bows.

The Traditional Bowyer's Bible series does however have chapters devoted to ethnic bows. These are very well researched chapters which go into great detail about use, origin, etc. These chapters will probably cover what you had in mind.

The best book on the English Longbow is co-authored by Matthew Strickland and Robert Hardy, The Great Warbow.

In the ~500 pages, you'll get a very comprehensive study of the bow used by the English. The appendix also has a very illuminating study which produced data for a 150 lbs longbow and several arrow types. It showed that elevation gave the archer an incredible advantage.

It is much harder to find good sources on the other bows. Adam Karpowicz is the authority on Turkish Bows. Read his article on the performance of Turkish Bows, here

It is maybe not that easy to see but the Turkish Bow was much more powerful. If you use the most efficient arrow for the heaviest bow that Adam tested, it produced 151 Joules.

Looking at the data from The Great Warbow, we can see that the Longbow at best produced 134 Joules.

The Turkish bow produced almost 13 % more kinetic energy. This may not seem all that much of a difference until we look at the draw lengths and draw weights used for each bow.

The Turkish Bow had a draw weight of 136 lbs and was drawn to 30 inches. The Longbow had a draw weight of 150 lbs and was drawn to 32 inches. Not only was the longbow 10% heavier but was drawn out 7% more than the Turkish bow. Both draw weight and draw length are directly proportional to elastic potential energy. The Turkish Bow could have expected a performance increase of these 17% in theory if it was made to the same specifications as the longbow.

Now, that covers English Longbows and the performance of Turkish bows. That is not very much at all, and it's sad but the other bow cultures didn't receive much attention. English language sources for them is sparse. For example, we simply don't know what draw weights the horse bows of the Mongols were. It's also hard to make assumptions because we don't have any archers today who could be said that they have reached some sort of human potential when it comes to draw weights on horseback. We can make these assumptions for the English Longbow because we have more than a hundred surviving period bows which according to mathematical models and reconstructions prove these high draw weights (average of 150 lbs, reaching into the low 200s) and because we have archers today who specialize in drawing these heavy bows (Joe Gibbs effortlessly shoots 170 lbs and Mark Stretton holds the record for shooting 200 lbs).

It's even harder to say for the Yumi because modern Kyudo is target archery with Zen influences. The draw weights are intentionally kept low. And unlike the Chinese, the Japanese did not leave written records of draw weights. Additionally, the shooting style of Yumi (at least today) produces higher arrow velocities from a snapping motion of the wrist which is very unique to Japanese archery. It's not possible at the moment to estimate how much the wrist snapping could add to arrow velocity but it's clearly beneficial.

Ancient bows are very baffling. For example, we have Xenophon's Anabasis which compared Greek archers to Persian archers and in turn to slingers. One could assume that Greek archers only managed a paltry distance of 150 m. I don't know of any surviving examples or accurate reconstruction of Roman or Greek bows. We are more lucky with Egyptian bows though.

It's often said that composite bows like those used in most of Asia originated because of a lack of quality bow wood. Personally, I am not impressed by that explanation.

Finally, you might want to take a look at Traditional Archery from Six Continents by Charles Grayson. It seems to be what fits your interest the most. I haven't read it yet, so I can't say for sure.