In 166 CE, a group of men claiming to be an embassy from the Roman emperor Marcus Aurelius arrived in the Chinese court at Luoyang. They weren't really an embassy - more than likely, they were just some marooned merchants from Alexandria - but they had managed the impressive feat of traveling the length of Asia. If some masochistic impulse had impelled them to make the journey overland, they could have followed the lacing branches of the Silk Road the whole way from Rome's eastern frontiers to the western marches of Han China. And over the five thousand miles of that wholly unnecessary journey, they would have crossed only three borders: from the Roman Empire to the Parthian, from the Parthian to the Kushan, and from the Kushan to the Chinese.
By the beginning of the third century CE, the political structure of Eurasia had been more or less stable for a long time. The border between the Romans and Parthians had shifted little over the years, though the Romans had managed to claw away bits of Mesopotamia and Armenia. The Parthian realm had gone through a few cycles of fragmentation and consolidation; but it retained more or less the same borders it had defended since the great Roman campaigns of the Late Republic. The Kushans - as far as we can tell - controlled much the same parts of northern India and central Asia that they had for centuries. And the Han Dynasty, though riven by rebellion and gradually receding from its conquests in the Tarim Basin, still governed virtually all of its traditional domains.
A half-century later, the world had changed. The Roman Empire was still there, but torn by civil war and battered by invasions. The Parthians were gone, replaced by the Sassanid Dynasty. The Kushan Empire had withdrawn to India. And China was divided between three kingdoms.
It is tempting to connect these stories of decline, and there certainly were ties between them (as we'll see shortly). But it would be misleading to assume that they had the same root causes.
To begin with the Romans: the half-century between the assassination of Severus Alexander (235) and the accession of Diocletian (284) is sometimes called the military anarchy. Whatever you call it, it was an unpleasant time to be Roman. Domestically, the chief problem was a crisis of imperial legitimacy. Emperor after emperor was proclaimed by the legions, none lasting long enough to establish a dynasty, each compelled to placate the troops with currency-ruining raises and bonuses. The domestic crisis was worsened, and largely driven, by developments beyond the frontier. To the north, unprecedentedly large and well-organized tribal confederations swept over the northern frontiers. And to the east, the shambolic Parthians were replaced by the aggressive and expansionist Sassanid Dynasty, whose kings Ardashir and Shapur led massive raids into Roman Syria.
The Parthians had never had a strongly unified realm. Part of the problem was the decentralized nature of their imperial administration, which seems to have devolved a great deal of power to local dynasts. External pressures were another perennial weakness. From the reign of Augustus to their third-century demise, the Parthians were Rome's punching bag. Their capital was repeatedly occupied by Roman troops, and thier dynastic politics were chronically complicated by Roman interference. The nomads on their northeastern frontier were an additional long-term hemorrhage on the treasury. In the end, however, the Parthian Empire was brought down from within. Ardashir, the founder of the Sassanid Dynasty, raised his rebellion in the wake of a protracted war between the Parthian king and the Romans, and seems to have taken advantage of the ruling dynasty's exhaustion. A battle was fought, the Parthian king fell, and his dynasty died with him.
We know less about the Kushans than we do about their neighbors. Their decline in the mid-third century, however, can be blamed largely on the Sassanids, who conquered Kushan territories west of the Indus River. After these campaigns, the Kushans withdrew to the southeast, where their control would endure until the rise of the Gupta Dynasty.
The Han Dynasty, finally, declined and fell for reasons that had nothing to do with their western neighbors. The seeds of its collapse had been planted centuries before, when Emperor Guangwu, restoring dynastic control after the traumatic rebellion of Wang Mang, established a regime that encouraged local landlords to become increasingly powerful and allowed the imperial court to become progressively more remote. A series of disasters in the late second century (most famously, the Yellow Turban Rebellion), combined with a series of weak or underage emperors (often controlled by court factions) to create a power vacuum exploited by a generation of over-mighty generals.
There were interconnections between these four stories of decline, at least to the extent that the rise of the Sassanids destabilized both the Roman and Kushan regimes. And one could postulate meta-narratives of decline, based, for example, on the lingering effects of the Antonine Plague (which probably swept across Asia in the late second century), or on long-term climatic changes. But even if such a link could be proven, the declines of all four empires would have to be understood on their own terms.
Edit: Several insightful commentators have suggested that climatic changes may have played a substantial role in the general unrest of the early third century. This is certainly plausible, and some evidence exists for a period of cooler and wetter weather. I do not, however, believe that there is enough data to make any conclusion about the influence of climate, particularly since the decline of each empire can be adequately explained by social and political factors.
Finally, I have to say that I've been a bit overwhelmed by the response to this post; thank you all for your comments, for your awards, and for taking the time to read!