The sources for pre-7th century battles lacks any detail and even sources of the 7th and 6th centuries BC are only sporadic so we can only guess.
The Neo Assyrians seem to have used cavalry as early as the 9th century BC in number, whereas cavalry didn't seem to be introduced into China until the 5th century BC. It's not until the late 8th century however, that cavalry seem to have gained a clear prominence over chariots. In this, the Assyrian armies (and many other Near Eastern armies) should be more mobile than the contemporary Chinese armies. However, the role of cavalry was not as aggressive as it would be in the 4th century BC where they would charge right into enemy formation. Tactically, the early cavalry usually only charged after missile fire weakened the enemy formation and they played a secondary role to the infantry (most of which were archers).
There are virtually no records in detail of how the Assyrian infantry fought, but if they were at all like the later Persian army, which were also prominently archers, then their formation is probably less sophisticated than those of China's. The Persian infantry were mostly archers and lacked compact formations and that's why they often got bested by the Greek hoplites, which maintained cohesive defensive formation.
The deployment of the Persian infantry usually consisted of a line of spearmen holding large shields at the front protecting a number of archers and missile troops behind. That is not to say that the Persian spearmen were not well armoured soldiers, who were highly skilled in individual combat. Persian spearmen, just as their Assyrian counterparts, were very able close-quarter warriors. However, very rarely were they required to fight in a close formation. Each soldier fought independently in the melee protecting his section of the shield wall supported by the archers and light infantry behind him. At the battle of Mycale Herodotus states (9.102), The Persians, as long as their line of shields remained intact, successfully repelled all attacks and had by no means the worst of things;...They [the Athenians] burst through the line of shields and fell upon the enemy in a mass assault. For a time, indeed, the assault was held, but in the end the Persians were forced to retreat within the protection of the barricade.The Persians were very able individual warriors at close quarters, especially protected behind their shield wall, but against an organized and disciplined mass of well-armoured heavy infantry their shortcomings could beexposed.Theconcept of individual heroism permeated the Persian military society. Nobles strived to excel at fighting as individuals with the bow, javelin and spear. The Immortals, an elite Persian unit of infantry, were well armed individuals who could fight effectively as archers or spearmen in close quarters. But they did not fight in any tactical formation...According to Herodotus, there is no differentiation between units and all Persians, that is all ethnically Persian soldiers, were armed in the same way. (CALGARY 2012)
Chinese infantry formations on the other hand, seem to be much more complex than even contemporary Greek hoplites and even adopted equal arms in proportion:
"Greek kill mechanisms were simple specifically because the form of the target, the enemy phalanx, was always constant and unchanging. Chinese kill mechanisms were much more complex because the disposition of the enemy always played a major role in determining how that enemy should be targeted. This relational aspect, not present in Greek warfare, greatly complicated the strategic situation, placing the onus on strategists to out-think their opponents rather than merely overwhelming them with direct force. Direct force was rarely adequate for achieving victory, since the goal became to understand the disposition of the enemy and be able to exploit it while concealing your true disposition from the enemy. This goal, when reduced to its lowest terms, is really a relatively simple optimization problem."
Michael Allers: CLASSICAL GREEK AND CLASSICAL CHINESE WARFARE: A COMPARATIVE ANALYSIS:
When the Greeks were struggling to escape the confining nature of the phalanx and its single tactic of the mass collision, China had already perfected numerous formations and methods of deployment, as well as an underlying hierarchical organization based upon the squad of five that, when coupled with precise training methods, allowed articulation, segmentation, and the execution of both orthodox and unorthodox tactics. -The Essence of War: Leadership and Strategy from the Chinese Military Classics, Ralph D. Sawyer p.1 Intro
However, this only pertains to the 6th century BC and after, details of battles in early China are rare, this is probably the earliest account; the battle of Cheng Pu in 632 BC.
The Qin center army consisted of two sections: the main infantry force making up the majority of the center army, and an elite force, Duke Wen's bodyguards, called the "first army," attached to the right flank of the main part of the center army. Qin attacked first, beginning with an advance of both left and right armies. The Qin left engaged first, smashing the Ch'u right army in an attack that was "urgent, impetuous, and rapidly successful" . The Qin right army then served as a "holding force, fixing the Ch'u center and preventing it from attacking the Qin center or aiding the Ch'u left wing". The Ch'u center could not move, since doing so would invite a flank attack from Qin left. While this was happening, Qin right was also advancing. They advanced to a predetermined spot, within bowshot range, and quickly reversed, feigning flight. The Ch'u left took the bait and pursued. Before the battle, tree branches had been pre-placed in front of the Qin right antly at the point where they turned and feigned flight. As the Ch'u left army made its way towards these branches and the fleeing Qin right, the Qin chariots swept across the front, dragging the tree branches. This dragging action caused dust to rise, obscuring the "fleeing" Qin right army, who, behind the veil of dust, was presumably circling out left with plans to take Ch'u's flank. The chariots did not engage the advancing Ch'u left army. Instead, as they approached, Qin's "first army," the Duke's elite unit, broke from the center and swept into the Ch'u left army flank. At this precise moment, the Qin right army who had feigned flight appeared at the scene to rout the Ch'u left army .
Thus, while exact force structures are not given, we can glean from the story that the major force structure themes were utilized. As for the Qin army, they were highly mobile-they were able to execute an effective feint involving a mass infantry force, as well as several sweeping and flanking maneuvers. Qin used combined arms in a most ingenious way as part of a ruse de guerre. Interestingly, as suggested by the theorists, the chariots played only a supporting role and did not engage infantry forces. The Qin forces must have been able to transition quickly between formations-the feigning maneuver and subsequent circling and attack demonstrates this. We also know from the Tso chuan and Shih Chi accounts that rigid hierarchies were in place and largely responsible for the effectiveness of the Qin army in routing Ch'u. On the other hand, the Ch'u army seemed rigid in their formations and their strategy. The Ch'u center stood its ground while an entire third ofthe Ch'u army was smashed.
The battle shows the state of Jin already had flexible formational changes, while the state of Chu were still relatively rigid in formation (but at least they had a cohesive formation). It also seem that infantry did not fight separately from chariot in China until the late 7th century BC. The Assyrians also used iron, whereas the Zhou used bronze, but unless some studies provide HV levels and qualitative comparison between the two, early iron is not necessarily superior to good bronze, especially when furnace temperature in China was the highest in the world, and by 600 BC, were already making bimetallic blades that were harder than iron.
In sum, Near Eastern armies had a lead in mobility because of their adoption of cavalry, whereas Chinese infantries tend to have more sophisticated and flexible infantry formations and had a more variety of arms (halberds, spear, crossbows, and bows). If Chinese armies before the 7th century BC were not like they were later, then Assyrian probably had a slight edge in quality at the time (especially considering it had infantries operating independently from chariots, and can fight in more variety of terrains). It's hard to say which army in the 7th century had the edge. After the 7th century BC however, given that infantry archers seem to still be the primary Near Eastern arm, I would give the lead to Chinese armies by then because even in the Near East, infantries were more decisive than cavalries. In the 4th century BC, cavalries became more aggressive and overtook infantry as the more decisive unit, and both the Persian and Macedonian cavalry would head on charge into missile firing and infantry formation without the need for missiles to soften them up, but by then Chinese infantries had more powerful crossbows with complex trigger mechanisms that often blasted off direct frontal heavy cavalry charges (and cavalry also became a unit in China itself).