I unfortunately do not have the details in my head anymore to properly answer your question, but perhaps you can find partial answers in my earlier posts on the subject. I link to two other posts here.
If you have access to academic journals, I can recommend the articles by Eduard Alofs. He has an interesting take on exactly your question and summarises the response to nomadic threats in the Middle East as the "Iranian Tradition", as opposed to the nomadic "Turanian" edition.
Alofs, Eduard, ‘Studies on Mounted Warfare in Asia I: Continuity and Change in Middle Eastern Warfare, c. CE 550-1350 – What Happened to the Horse Archer?’, War in History 21 4 (2014) 423-444.
Idem, 'Studies on Mounted Warfare in Asia II: The Iranian Tradition – The Armoured Horse Archer in de Middle East, c. CE 550-1350', in War in History 22 1 (2015) 4-27.
Idem, ‘Studies on Mounted Warfare in Asia III: The Iranian Tradition – Cavalry Equipment, Infantry, and Servants, c. CE 550-1350’, War in History 22 2 (2015) 132-154.
Idem, ‘Studies on Mounted Warfare in Asia IV: The Turanian Tradition – The Horse Archers of Inner Asia, c. CE 550-1350’, War in History 22 3 (2015) 274-297.
The settled neighbours of steppe groups using mounted archery often used mounted archery themselves. Thus, we see Chinese, Russian, Byzantine, Indian, and Persian armies with mounted archers. Sometimes such states obtained by mounted archers by hiring steppe nomads as mercenaries, and sometimes they trained their own mounted archers. In the latter case, they were not always as skillful as the steppe archers (e.g., Chinese writers complaining that (mounted) archers only practiced shooting to the left, and ignored shooting ahead and behind since those skill weren't examined).
The other solutions to steppe mounted archer employed by neighbours included fortifications, crossbows, and guns. In particular, crossbows and guns can both outrange mounted archery, or at least keep enemy mounted archers at extreme range, thereby lessening their danger in battle.
The basic design of the Asian composite bow remained the same until after the Mongol conquests. There were regional and temporal differences in style, in the type of covering for the limbs, whether the upper and lower limbs were the same length, whether the ears/siyahs of the strung bow were angled forwards (requiring a string bridge/pad at their base) or angled backwards, and whether there were horn or bone reinforcements at the handle and ears. Two additional changes should be noted:
There might have been an increase in average draw weight from the earliest bows to later bows, as armour became more common (followed by a decrease in draw weights as armour disappeared from the battlefield as guns became more common). For Ottoman military bow draw weights, see A. Karpowicz, “Ottoman bows — an assessment of draw weight, performance and tactical use,” Antiquity, vol. 81, pp. 675–685, 2007. These average 120lb. Of similar or higher draw weight is the Tsagaan Khad bow, a 14th century Mongol bow, discussed in https://www.reddit.com/r/AskHistorians/comments/928g79/it_is_said_mongol_bows_draw_weight_was_as_high_as/
The Manchu bow was a very large composite bow, with very long siyahs. Unlike the typical Asian composite bow, which was used to shoot arrows of about 30g, the Manchu bow was used with very heavy arrows (e.g., 100-125g). With the Qing conquest of China, this bow became the standard military bow in China, and influenced bow designs through much of northern Asia. For example, a shorter-eared version was adopted in Mongolia, and was used there into modern times, and from Mongolia, this design spread west, appearing as far west as the Crimea. The Ottomans developed a bow intermediate between this Mongolian bow and the traditional shorter composite bow, calling this new bow a Crimean bow. For examples of these newer types of bows, and examples of the older style (which still remained in use), see Charles E. Grayson, Mary French, Michael J. O'Brien, Traditional Archery from Six Continents, University of Missouri Press, 2007.