I suppose you're referring to the Emishi who lived in mainland Japan (namely Tohoku) before they were conquered. The Ainu strictly refer to tribes living in Hokkaido, which was not annexed by the Japanese until the Meiji Era, when the Empire of Japan came rolling in with rifles. The Emishi, on the other hand, were the aboriginals of Honshu before the Yamato Empire started conquering it bit by bit until they achieved full dominance around the 9th Century. The conservative theory is that Ainu are Emishi who have survived by being in Hokkaido (Ezo at the time), where the Yamato Empire did not expand. This would make Ainu "modern day" Emishi. However, since there is academic dispute concerning the link between Emishi and Ainu, and because by definition Ainu refers only to the indigenous people of Hokkaido, Sankhalin and the Kurils, I still feel the need to make a distinction. Since you're talking about the Yamato Empire, who never made it to Hokkaido, I'm going to assume you're talking about the Emishi.

It's widely hypothesized that Emishi are the descendants of the Jomon people. The Jomon people had been living in Japan ever since the Ice Age, possibly having crossed when an ice bridge was formed to the archipelago. They were a hunter gatherer society, with signs of early agriculture, who got by hunting with the bow, catching fish and collecting nuts. With no external stimuli and relatively unfettered reign over Japan, they stayed that way for a long time.

Then came the Yayoi people. These came from mainland Asia, some time between 10th-3rd Century BC, and landed in Kyushu, to the southwest of Japan. They brought with them technologies the likes of which the Jomon had not seen before. They brought with them metallurgy in the ability to smelt and create iron tools. They brought with them true agriculture in the form of rice paddies. This technologically superior civilization spread very effectively, either assimilating their Jomon competitors, or outright eliminating them (most likely a mix of both). The Yayoi people expanded from Kyushu all the way to Kansai in Honshu.

Very little is known about why the Yayoi stopped there (or if they did) since they had no writing system of their own. What little writing we have on them is from Wei China, who records that they were plagued with war between Yayoi kings. Ultimately, this infighting stopped when the Yayoi were unified. Whilst the details are unclear and shrouded in myth, eventually they would united under a single Emperor, becoming the Yamato Empire.

Fast forward to the Nara period (8th Century AD). The Yayoi-Jomon rivalry survived in the form of the Yamato-Emishi conflicts. The Yamato ruled the western half of Japan from their capital at Yamato (modern day Nara) whereas the Emishi thrived in the untouched Eastern half. However, both sides had evolved significantly since then. The Yamato people, which were a mix of Yayoi and the Jomon that they assimilated still held the technological advantage over the aboriginals. However, there technology has gone a far way, having assimilated technologies from the Chinese. They now had Chinese-style writing, clothing, architecture and most pertinently to this case, soldiers. The Yamato Empire had developed a standing national army. Imitating the Chinese, they had lamellar armour, and were armed with iron swords and rectangular shields. The Emishi, on the other hand, had assimilated some of the technologies of their neighbours, but the most important change came in their warfare.

The Emishi were marked by being predominantly horse archers. They were experts at rushing forth to battle on rapid steeds, firing shots, and then falling back. Horse riding was a great part of Emishi culture. This is one of the reasons argued against Ainu having descended from Emishi, as the latter do not inherit these traits.

The Emishi were generally subservient to the Yamato, recognizing their technological prowess and agricultural might to be superior. However, in the Nara period, there was a rebellion against the Yamato Emperor and Yamato marched its national army in a punitive campaign against the Emishi. The campaign ended in disaster.

The Yamato were very ill-suited to fight the Emishi, which is likely why they never took over Tohoku in proper. The Yamato army, with its soldiers sporting large and heavy shields, moved slowly. This cumbersome army had difficulty dealing with the hit and run tactics of the Emishi. The Emishi could easy outflank them, hit them with a volley of arrow, and retreat on horseback. The horse archers were able strike the shielded soldiers with impunity whilst the latter could not catch up to their assailants. Despite their numerical superiority, the Yamato forces faced a crippling defeat at the face of their enemies.

This defeat had huge repercussions in the Yamato Empire. Their state-of-the-art, numerically superior army that they sent to destroy a foe that they viewed as inferior barbarians were, on the contrary, defeated. Faith in the Empire and its national army dwindled. The Emperor at the time, Emperor Kanmu, burdened with the task of surmounting this obstacle, had to turn to other ideas.

The idea he implemented was to essentially fight fire with fire. In this case, it would be horse archer with horse archer. Yamato was to raise an army of horse archers of his own, but there were no such people in the national army.

Near the frontiers of the Yamato Empire, in the expanses of the Kanto plains, there lived a group of people who, due to proximity with the aboriginals, had developed a similar fighting style. They were different than the civilized folk living in Nara. They lived in clans, rode across the plains on horseback, wore face paint like their Yayoi ancestors but, most importantly, they were horse archers. They were Yamato, yet they lived like the Emishi. Emperor Kanmu appointed a leader out of them, Sakanoue no Tamuramaro, and bestowed upon him the rank of sei-tai-shogun. This rank, likely created for him, was an emergency title giving a person control over military affairs until a crisis is averted. The Yamato Empire had Sakanoue gather the clansmen of the Kanto plains and welcomed them into its service, giving them the name of "Samurai" (those who serve).

The samurai horse archers were very adept at fighting the Emishi. They had the same background of horse archery yet they were armed with the armour and equipment of the Yamato. They were supplied by the Yamato Empire and did not lack in food or ammunition. The various samurai clans rode across the East and put down the Emishi rebellion.

After having shared Japan with aboriginals since their arrival at the islands, the descendants of the Yayoi finally defeated the descendants of the Jomon and claimed Eastern Japan as their own. Faith was lost in the national army, whereas the samurai clans had gained the favour of Yamato nobility. Samurai clans became the preferred military force, causing the national army to disband. The dependence on samurai and the disbanding of the national army, caused a massive decentralization of power to occur. Power left the imperial court and flowed into the hands of regional samurai clans. In the following Heian period, Japan saw samurai rise to become the defacto rulers of the nation, with the Emperor reduced to a religious figure. This decentralization also led to each clan's domain becoming a state of its own, marking the death of the Yamato Empire and the start of Feudal Japan.