The Assyrian and Persian Empires were both large, multicultural, and multi-ethnic empires of the ancient world. The Assyrians rose to dominance first, collapsing by the end of the seventh century BCE. Their empire would, at the its height, extend across Mesopotamia and the Middle East as far west as Egypt. Persia's greatest period of expansion, on the other hand, would come under Cyrus the Great (who would die around 530 BCE). The Persian Empire was much larger than the Assyrian Empire (and history's largest empire up to that time), conquering much of the same territory that the Assyrians had, but extending further west across Asia Minor and further east towards the Indus River.
Both the Persians and Assyrians were monarchies, which ruled with the assistance of elaborate systems of administration (to more effectively govern these far-flung, multi-cultural empires). As historian D. Brendan Neagle writes about the Assyrians:
The lands subdued by his campaigns were turned into provinces administered from Assyria by a governor directly responsible to the king. In Assyria itself existing administrative divisions were reduced in size to improve central control. An efficient communication system was set up using messengers who carried reports back and forth from the governors to the central administration (The Ancient World: A Social and Cultural History (5th Ed). Upper Saddle River, New Jersey: Prentice Hall, 2002, p. 63).
When looking at this description, you can certainly see parallels between the Assyrian and Persian systems of administration. Later, the Persians would subdivide their empire into the various satrapies, while also relying heavily on internal communications, tying their empire together by the use of road-building.
Additionally, both empires had a great deal of internal instability, which rulers had to reckon with. To again quote Neagle,
When Babylon revolted in 689 BC, Sennacherib (704-681 BC) destroyed it....In 671 B.C. Egypt was subdued but quickly rebelled and had to be beaten down again eight years later. On the death of Ashurbanipal in 627 BC, a dynastic struggle racked the Assyrian state. Babylon again revolted, and under the Aramaean (or Chaldaean) Nabopolassar (625–605 BC) made common cause with the Medes of Iran and bands of Scythian nomads from Eurasia...In the west Palestine was again in revolt. In 614 BC Assur fell, and two years later Nineveh was captured. By 609 BC the remnants of the Assyrian army were being stamped out in northern Mesopotamia (65)
Similarly, the Persians also had to reckon with a significant degree of internal instability. Among the most historically famous revolts was the Ionian Revolt by the Greek cities in Asia Minor, which famously precipitated the Persian Wars. As a second example, you can point towards Egypt, which successfully drove out the Persians in 404 BCE (only to be later reconquered approximately sixty years later, shortly before the conquests of Alexander the Great).