As in, the Mongol tradition of horseback archery goes back centuries, but was this tradition forgotten, with the modern versions displayed at Naadam festivals being a resurrected version, not something passed down generations?
Or maybe it evolved into other continuous traditions, through its relationships to Manchu archery, Chinese archery etc.
Modern Mongolian mounted archery is largely, if not completely, a reconstruction. Mounted archery survived into the 20th century in Mongolia, but was in serious decline due to greater availability of guns, which had replaced the bow as the main weapon for hunting as well as for warfare. Mongolian archery experienced a renaissance with independence from China and revival of the Naadam festival. However, the archery competitions at Naadam are on foot, so mounted archery continued to decline.
During the 20th century, Mongolian archery has evolved, with the arrow usually being shot from the left side of the bow (by a right-handed archer) even though a thumb draw is used (for a right-handed archer, the arrow is usually on the right of the bow with a thumb draw, and on the left for a finger draw). The arrow is often nocked above the drawing hand, rather than secured between finger and thumb (this traditional method is still used, too). Shorter draws are often used. Modern bows are sometimes used, sometimes made in the traditional form (but with fibreglass limbs), and sometimes modern Western-style bows.
In the mid-20th century, Mongolian archers were still shooting in the old style (arrow on right, long draw):
so these changes have mostly been in the second half of the 20th century.
Mongolian mounted archery is currently undergoing a revival, driven in part by the appearance of a worldwide competitive scene for mounted archery. However, with most of the last old traditional mounted archers being old already in the early 20th century - people who had learned in their youth in the 19th century when bows were still widely used for hunting - the 21st century revival is more reconstruction rather than a continuous tradition from the old days.
Mounted archery survived for longer in Tibet, with Tibetan cavalry still carrying both bow and muzzle-loading musket in the mid-20th century. However, Tibetan archery competitions at festivals are on foot, and mounted archery in Tibet appears to have gone the way of Mongolian mounted archery (complete with something of a modern reconstruction and revival). Some photos of traditionally-equipped Tibetan cavalry can be seen at http://www.manchuarchery.org/photographs-tibetan-archers
In Japan, mounted archery (yabusame), has survived as a continuous tradition, due to its use as a ritual in religious festivals:
This gave the tradition a reason to persist even though it was long obsolete for hunting and war. Of late, the ritual use has given rise to mounted archery as a sport in Japan:
which could result in wider popularity.
Some modern Mongolian mounted archery in action:
The shooting at the shields carried/worn by another rider is definitely a "do not try this at home" kind of thing!