They expanded the borders and incorporated the foreigners not just into their populations but their armies as well; in the end the Roman Empire collapsed because too few of its people were “Roman”.
As Cronos pointed out (as have others, as well as yours truly), the Roman Empire went on until the Turks took Constantinople, and they were very interested in the learning of classical antiquity, as well as the imperial trappings, and saw themselves as the successors to earlier rulers (other than their choice of religion of course).
Rome started its conquests by taking Ostia (now Poor Man's Pompeii, but with the virtue of being an easy distance from the popular Roman tourist attractions in the old urban core), meaning the Empire started much earlier than is sometimes claimed, and continued into the 15th century, a total of nearly 1900 years. Yeah, that opening up citizenship to people from outside the citi really didn't work out, did it? The first emperor born in one of the provinces was Trajan, who began his reign in 98 AD, and he was one of the most successful, conquering the future birthplace of Aurelian, who is a favorite of mine. :^)
The use of auxiliaries began at least by the time of Augustus (generally regarded as the first emperor) who cut the regular army in half after the defeat of Anthony and Cleo, to 28 legions, and added the Praetorian Guard (emperor's bodyguards and the local police force) and 28 legions' worth of auxiliaries.
One ancient writer joked that he had to travel into the provinces to hear Latin spoken, as so many speakers of other languages had crowded into Rome; Ovid by contrast was exiled to a largely Scythian area on the north shore of the Black Sea, and wound up thinking, dreaming, and writing in the local tongue (by his own account).
If anything, the Romans were pathogically suspiicious of foreign peoples in groups -- yet they welcomed their cults into the capital and other cities (the rites of Mithras were common among Roman soldiers, so much so Mithras was called "the soldiers' god"). Barbarian tribes were considered inferior, yet were sometimes added into the Empire via conquest of new territories. The view that the Varian disaster was "The Battle That Stopped Rome" was ridiculous when it was promulgated, and looks worse with each new discovery; just in the past ten years a Roman cemetery was dug up in Copenhagen Denmark, and Maximinus Thrax was winning battles to the south of there, deep in Germany, and he had a short reign -- the Guard assassinated him during a Senatorial revolt brought on by his so-called low birth. He was as non-Roman in ancestry as could be, but was diligently adding to the Empire. His sudden removal led to the 3rd century anarchy which lasted about 70 years, and yet, surprisingly, the Empire survived that (thanks to Aurelian, Diocletian, and others).
Among the reasons for the decline of Roman power I'd list, lack of public education and/or general literacy (thus, lack of a common history and mythology); dilution of Senate authority on ridiculous acts, such as voting dead emperors and other deceased Romans into the company of the gods; sketchy banking system (all of it was private lending, similar to what the Medicis and others were doing during the late Middle Ages and into the Renaissance); no postal system worthy of the name; and most importantly, no statutory and regularized system for succession into political power for the Emperors (reform of that didn't arrive until Diocletian).