And, of those examples, it was when the states became more monocultural that they experienced stagnation followed by a severe decline.
Not only were the Romans, Persians, and Muslims minorities in their Empires, but small minorities. They thrived on new ideas, and expanded. In the later portions, the culture became more uniform, the vast majority assimilated into the culture, and progress slowed. Xenophobia also grew, and this led to great political divides because you had different groups claiming "We're the TRUE X culture," at the same time as attacking people living in their own Empires they deemed as alien. Why do cultures become foolishly xenophobic? Largely it's because Xenophobia is easy to spread among commoners, and xenophobic people are very easy to manipulate because the mere suggestion that someone is invading them is enough for them to submit to any leader who claims they will protect them... this stage usually occurs near a civilization's collapse, or on the verge of a catastrophic defeat - history rarely proves kind to xenophobic nations/empires, no matter how powerful they appear; they are always at a disadvantage.
In short, those Empires which manage to limit xenophobia thrive, and see golden ages occur. Those who suffer from it tend to fall into stagnation, decline, and conflict with other cultures internally.