Were composite bows effective on the post-gunpowder battlefield?

While the reflex-recurve composite bow does have better energy storage for a given draw weight (and draw length) than a longbow, and typically higher efficiency with light arrows, "far better than European longbows" seems exaggerated. About 20% more stored energy is fairly typical, and poor efficiency of a longbow with light arrows is readily solved by using heavy arrows (and large composite bows with heavy limbs, such as the Manchu/Qing bow, have the same efficiency issue, as do other big heavy bows like the Japanese yumi). The composite bow is better, not considering time needed to make them, expense, etc., but it isn't a super-bow compared to longbows. More on this in my reply to https://www.reddit.com/r/AskHistorians/comments/ba9yga/why_did_late_medieval_europeans_namely_the/

In multiple cases, when mature smoothbore guns were available (e.g., the matchlock musket or arquebus, as opposed to older hand-cannons), they quickly displaced the bow as the infantry ranged weapon of choice. Since the evolution from the hand-cannon to the arquebus took place over time, in some parts of the world, the bow and gun were used side-by-side as infantry weapons for a long time (e.g., in China and much of Europe). Where earlier guns hadn't reached, or hadn't been adopted in large numbers, the superiority of the arquebus/musket often led to its enthusiastic adoption (Japan, North America, sub-Saharan Africa, New Zealand (although in this case, without an older archery tradition to compete with), etc.). So there was clear recognition that the musket/arquebus was a powerful and effective weapon.

Given the rate of fire and range of bows, why was this so? The importance of range is often exaggerated. While light arrows can be shot long distances with the composite bow (except large heavy ones such as the Manchu/Qing bow), there remains the problem of (a) hitting the target and (b) doing significant damage to the target. While the composite bow, if Ottoman flight archery records can be believed, can shoot an arrow over 700m, it is a quite different matter to hit a human-sized target at that range.

As far as doing damage to the target, note that kinetic energy is lost through drag as the arrow travels to its target, and long range shooting is unlikely to penetrate armour, even light armour. Even at point-blank range, much armour was sufficient to stop arrows - stopping arrows was a major goal of armour, and breastplate and helmets, covering the body parts where arrow wounds were most dangerous, were often thick enough to keep arrows out. The lighter armour on arms and legs might not stop arrows at short range, but would have been effective at long range. The importance of close-range high-power shooting to try to defeat armour contributed to adoptions of large heavy bows (the Manchu/Qing bow, the Japanese yumi), at the cost of long-range performance. Large heavy bows have an additional military advantage over the lightweight efficient forms of the composite bow: they are usually more damage resistant. Durability can matter on the battlefield.

Trying to defeat armour was the key factor that pushed military draw bow weights so high (130lb and up for longbows, 100lb and up for composite bows). But very high draw weights have their disadvantages - they make training archers more difficult, and if archers are sick, hungry, or exhausted (as soldiers might be) they might not be able to properly use their bows.

Guns capable of penetrating armour that keeps arrows out were an attractive weapon on the armour-rich battlefield. The effective range of a smoothbore musket was easily as great as the ranges over which a bow could be expected to be effective against armour. This brings us to one important function of early guns: defeating armour. If one considers the relative non-lethality of arrow wounds to the arms and legs (compared to the torso, neck, and head), getting through breastplate and helmets is even more important for bows. Less important for guns, since musket wounds to the arms and legs could be thoroughly incapacitating, with bones shattered, and limbs close to ripped off. For a discussion of the lethality of arrow wounds, see the comments by myself and u/FlavivsAetivs and u/zeeblecroid in https://www.reddit.com/r/AskHistorians/comments/9r1oed/how_were_archers_used_in_ancient_and_medieval/ The quick summary is that if the arrow penetrates the thoracic or abdominal cavity, or penetrates the skull, the wound will be fatal, either immediately or later from infection, and if the arrow hits an arm or leg, and doesn't hit a major artery (which can happen, but usually didn't) the wound would probably be survived, and might not be incapacitating (19th century rates of death from infection for such wounds were 5-10%).

Thus, important military advantages of guns were better performance against armour and greater lethality.

However, the musket has a major disadvantage: it is difficult to reload on horseback. The European solution was to use shorter guns as cavalry weapons: the pistol. However, the pistol is still very difficult to reload on a running horse, and lacks the power of the musket (with worse performance against armour, and less lethality). Where the composite bow was retained as a major weapon after the adoption of the musket, it was typically used as a cavalry weapon. In some cases, Asian cavalry carried both musket and bow, allowing the bow to be used after the musket was fired, bypassing the problem of reloading the musket in battle while on horseback. With the infantry using the musket, the infantry provided excellent anti-armour capability, and the cavalry could use lower draw weights. This was even more the case if the bow was used as back-up to a musket.

In these circumstances, the bow could still be effective. However, for cavalry to be effective, whether using bows or not, leadership, discipline, and sensible tactics are still required - all three appear to have been lacking in the Bashkir archers you note. Their bows were not as dangerous as the muskets the French were accustomed to facing, but the real issue appears to have been the lack of discipline and poor tactics. The Chinese successfully used bow-armed cavalry alongside infantry musketeers, and the bow appears to have still been useful. Even in the hands of infantry, the bow could still be useful against smoothbore muskets, and was used alongside muskets by infantry in India and North America. However, more modern guns made the bow thoroughly obsolete in the mid-19th century, and even the Chinese dispensed with it, as discussed by u/EnclavedMicrostate in https://www.reddit.com/r/AskHistorians/comments/b0b1ov/when_did_the_qing_dynasty_stop_using_swords_and/