Suppose stirrups (and the corresponding improvements in saddlery) are developed somewhat earlier. What might the consequences be if:
Given that stirrups may be the most overrated technological development in history, probably not as much as people think.
A. Some bright Roman comes up with them around 1 AD?
Riding may become a more popular mode of travel among the upper classes. In the medium term, this could affect how the Roman world's infrastructure works, but I don't think it will do so decisively. MIlitary applications will take a while because Rome has access to cavalry recruits from traditional backgrounds who will take time adopting the stirrup, but it might get popular faster in the cohortes equitatae and maybe some alae CR. Not likely big changes in the composition of the military or its application, but maybe different recruitment patterns in the medium term.
B. Some bright Greek/Persian comes up with them around 500 BC?
This would be a lot bigger because stirrups require a firm saddle of some kind. Give the Greeks saddles and stirrups and you will see a much more effective caalry arm. Thessaly might come to play a more important role in Greek affairs. Quite possibly the symbolic importance of the hoplite may be undermined. I can imagine especially in Magna Graecia the mounted aristocrats taking on a much more important role in warfare. To Persia, I can't say, but I suspect not too much of a difference.
C. Some bright Assyrian comes up with them around 1000 BC?
Also very interesting - it may get us earlier cavalry as a major arm and thus diminish the importance of chariotry. Different priorities in horsebreeding and training, and changed military budgets. With an earlier deelopment of shock cavalry (which is not a given), you might also see different infantry tactics. In the long run, this could undermine the perception of cavalry as a decisive arm.
D. Some bright Hittite comes up with them around 1500 BC?
E. Some bright Indo-European comes up with them around 2000 BC?
Actually, I doubt the chariot horses of that era would be up to cavalry use.