Did the knights of medieval Europe study the battle techniques of other cultures? Either ancient such as the Romans, Greeks or Persians; or concurrent at the time such as Chinese, Arabian or African.
Yes they did! The most famous example of this was Vegetius De Rei Militeri, which was very popular. By the 16th century there were translations of Vegetius available in Anglo-Norman, French, English, Scots, Castilian, Tuscan and German. The first French translation, by John de Meun, appeared in 1284.
Christopher Allmand argues that Vegetius was so popular, not just because of the general culture's reverence for the ancient world, or even contemporary military culture specifically, but because it was both informative and didactic. Allmand also points out that medieval translators would likely have been unaware of the specific circumstances behind it's composition; Vegetius was writing in the late 4th century when the ability of Rome to effectively project it's power beyond it's borders had weakened in comparison to its 1st and 2nd century AD peak. Vegetius then was arguing for military reform to return to that period of greater military strength. The medieval reader would likely not have seen that; just a reflection of ancient Roman military strength and wisdom.
Vegetius came to be used extensively by writers looking to create manuals for ideal princely behaviour, in an attempt to show the value of armies and proper leadership. Therefore he was used less for tactical advice, and more for advice on the proper preparations for war. Vegetius' recommendations were acted upon; Henry V acted in ways that resembled Vegetius' advice at the siege of Rouen in 1418/19. Vegetius advice may also have inspired Thomas Malory's portrayal of Arthur and his knights in Le Mort d'Arthur.
Further reading:
C. Allmand, 'Fifteenth Century Versiosn of Vegetius' De Re Militeri' in, M. Strickland (ed.) In Armies, Chivalry and Warfare in Medieval Britain and France (Stamford, 1998).
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