The theory comes up in particular with regards to the 19th century USA, where the westward expansion purportedly provided alternatives to the poor of the East Coast, leading to increased economic prosperity and/or reduced social tensions. It has also been mentioned as an explanation why labor union formation in the USA was limited as opposed to Europe where such opportunities were harder to come by.
This is an article about the issue (1935) to get you started.
Alternatively I'm interested in the application of similar mechanisms in other regions, like the other large expansonist semiperipheral state to Europe, the Russian Empire, or even the eastward expansion of European, in particular German, settlers throughout the Middle Ages, or perhaps the larger freshly independent Latin American ex-colonies which may have had similar opportunities for internal colonization.
The Roman Empire you say?
Oh, you said the Russian Empire. Anyway, there are actually some parallels with the Late Republic, in a way. Roman intellectuals often conceptualized the expansion of their empire as defensive and civilizing and it would be hard to make a case that it was driven by any real "pressure valve" mechanism, but the empire once attained did often have that characteristic. In particular, new imperial territory was often used to settle the "land question" (how to provide veterans with a land pension) in a way that did not deprive Italians. I can go into this a bit more if you are curious but it doesn't really exactly map onto the thesis as it applies to American history.