When we could go outside I was a budding amateur archer, and what surprised me most about the experience was getting to try bows from different archery traditions. Using a horse bow, long bow or traditional Japanese bow all felt like distinct and unique experiences with a different set of ideals and approaches. How much of an opportunity would there have been for crossover? Are there examples of Japanese or European archers coming across Mongol horse archers and riffing on/incorporating what they came up against?
In many cases it was a matter of geography: was a state or people isolated or did they exist in a region that facilitation transmission and interaction? This was a factor that strongly determined the spread of archery techniques and technology. The Eastern Roman Empire occupied a position that allowed it to be influenced by West European, Steppe, and Near Eastern cultures. An East Roman military manual written in the 6th Century AD, Maruice's Strategikon, states about a soldier:
'He should be trained to shoot rapidly on foot, either in the Roman or the Persian manner.'
So right there we have pretty clear evidence of crossover. Two different techniques were known to the East Romans, and both were practiced. The text also makes it clear that the East Romans adopted a lot equipment from a Steppe people know as the Avars. This includes lances, neck pieces, and tunics.
Earlier Roman artistic and written sources also show cavalry acting as close-combat troops. Trajan's Column, from the 1st Century AD, shows the an image of a Dacian cavalryman using a bow, but the Romans have their arms in a pose that indicates the use of spears:
http://www.trajans-column.org/?attachment_id=538
Several hundred years later, the Strategikon describes Roman cavalry as predominantly archers. This is another example of crossover, as the Romans came to adopt mounted archery practices in order to contend with enemies who also fought in such a manner.
Sources
Maurice's Strategikon: Handbook of Byzantine Military Strategy, translated by George T Dennis, University of Pennsylvania Press, 1984.
This is perhaps more evident when we look at regions where there would be significant intermingling of cultures, especially through warfare. One example is in Ancient Greece.
According to a research study, the depictions of archery techniques on surviving artwork was examined and compared. The findings were:
The Greek draw was a pinch draw, with the archer holding the arrow by the shaft and pushing it onto the string, pulling it to chest level. The reverse draw is shown as a inverted two-finger grip, believed to be inspired by the Scythians. Though the thumb draw was not commonly shown, thumb rings have been found from the Mycenaean period.
Source: Spyros Bakas, An insight view of the shooting methods of the archersf the Ancient Greek World 1400 BC - 400 BC
Also, the archery sources of the Middle East (Arab Archery and Saracen Archery) make references to other styles. From Arab Archery:
The Slavs (al-Ṣaqālibah) have a peculiar draw which consists of locking the little finger, the ring finger, and the middle finger on the string, holding the index finger outstretched along the arrow, and completely ignoring the thumb. They also make for their fingers finger tips of gold, silver, copper, and iron, and draw with the bow upright.
The Greeks have a draw which consists of locking the four fingers-the index finger, the middle finger, the ring finger, and the little finger-while the bow is in a horizontal position (rāqidah) ; and, holding the arrow between the middle finger and the ring finger, drawing toward the chest. This is indeed a corrupt draw used by the ignorant.
The thumb draw is the predominant method in Asiatic archery, but we do see from the text that the author was aware of other methods used elsewhere (though his opinion of the Greek draw was, perhaps, not so high).
The ultimate question of whether there was evidence of direct crossover and inspiration is probably one that cannot be easily answered. So much of archery was commonly shared and developed concurrently that we can't really track exactly where specific methods came from. Thumb draw was used long before the Arabs and Turks wrote down their manuals, and long before Gao Ying's 15th century manuals on Ming archery. We can tell that there is a difference in style and training from these sources, but they don't directly say where the methods came from.
The clearest examples of adoption of outside methods were the Romans, who made use of eastern archery styles after encountering the Parthians and Persians, as outlined in Strategikon.