A conversation I've had with Adrian Goldsworthy and Adrian Murdoch is about the introduction of stirrups in the West. The Greeks didn't have them, nor did the Romans, and this must have affected their ability to ride. The Chinese developed the stirrup, which did not appear until the 6th century in Europe.
For example, the horses at Luxor, which was an Imperial cult with frescoes from the time of the Tertrarchs, were saddled but had no stirrups (here; image):
This may be one of the earliest excavated examples, from an Avar grave in Hungary, where people from Mongolia emigrated and settled (source):
Excavations in Mongolia have confirmed that as their origin through similar finds there. The Avars settled in Hungary in AD 567, and we know they had contact with Justinian II at Constantinople by 560. The Avars proved to be formidable foes, so the Byzantines soon adopted the stirrup as noted in the Strategicon of Maurice c. AD 600 (skala; 1.2 and 2.9). Stirrups are first certainly shown in Western art on the Madara Rider in Bulgaria, dated by an inscription to AD 710 (source):
There are several other images of stirrups in use in the west in the 8th century. This fragment of Byzantine silk shows an emperor hunting; it was given to the Abbey in Mozac for the relics of St Austremoine by King Pippin the Short in AD761, providing a terminus ante quem (source p. 28 and Lyons):
The stirrup is shown in a Lombard Capitulare Evangeliorum, and was known by the late Merovingians and Carolingians, although it seems to have been used only by the elite. A Coptic ivory set into the ambo of Henry II (AD 1002) at Aachen shows stirrups in use by the Byzantine emperor ruler - it is generally dated to the 6th century which would make them the earliest depiction of stirrups if this is correct:
The frescoes from Qasr al-Hayr al-Gharbi built by the Umayyad caliph Hisham ibn Abd al-Malik around AD 727 show that they were using stirrups in Syria (source):
Although stirrups were not used by the Sassanians, they are shown on in Kushan art from the 2nd century BC onwards (seal of Adsho, BM; source):
What's interesting is that the Kushan / Indian stirrups or leather 'toe stirrups' did not catch on. Similar leather slings used for holding the feet are shown in art from the Steppes, such as the Chertomlyk Amphora (Hermitage), but again did not enter general use amongst their neighbours.
We know that the Franks were aware of stirrups, but initially did not use them widely. In AD 732 Charles Martel and his men fought on foot against the mounted Muslims at the Battle of Poitiers. By the Battle of Louvain in AD 891 the Franks were fully mounted, using stirrups, and this is how they routed the Vikings there. When and why they chose to adopt the stirrup is not known - the Alemanni whom the Franks absorbed had used them for some time - but this is when it became the norm in the West.
The use of the stirrup sometimes seems to be a personal choice - for example in scenes from the Apocalypse of Valenciennes some riders use them whilst others do not (dated by some to c AD 800, others to 900: source):