I'm currently reading a textbook on early US history, and came across the Turner Thesis - that the frontier fundamentally shaped the American psyche in a way different to any other country. I realise that it is far from perfect, but one thing struck me. Turner (I believe) called the frontier an "escape-valve", a means for the nervous energy and internal strife of the Eastern states to be directed externally, be it seeking the life of a cowboy and similar, settling a homestead or fighting Indians. Being more of a Mediaevalist myself, this made me think of Timothy Reuter's theory that the decline of the Frankish Carolingian Empire after the death of Charlemagne was caused by there being nobody external to fight; similar, perhaps, to Alexander weeping because there was "nowhere left to conquer". This was because annual campaigns to expand Francia ended, which had provided exercise for the Frankish warriors and a source of plunder that the king could use as patronage - both keeping the nobles quiet. The end of these external campaigns, segueing into less exciting and profitable border defence, led to rebellion, civil war, and decline. I know that the American Civil War certainly wasn't caused by the end of the frontier, and that no similar decline took place, but were there consequences of there being no more land to expand into, and if not, why not?
The debate over Turner has gone on for well over a century now, and a great many seminars have been taught around the themes he developed in his original essay and then repeated endlessly in everything he wrote or said in public between 1893 and his death in 1932. He was very much a one trick pony, at least in the eyes of his critics; if you add the idea of "section" to his frontier work you've pretty much summed up his entire intellectual output. But he has also had his defenders, which generally come in waves broken by periods in which his critics excoriated him for the assumptions and omissions in his work. If you're interested in his story and the endless debate over his thesis, I'd recommend Alan Bogue's 1998 biography [Frederick Jackson Turner: Strange Roads Going Down] (https://muse.jhu.edu/article/1698) as a starting point.
Representative examples of Turner's critics from various eras include the following:
Craven “Frederick Jackson Turner, Historian” The Wisconsin Magazine of History (June 1942)
Pomeroy “Toward a Reorientation of Western History” Continuity and Environment” Mississippi Valley Historical Review (March 1955)
Jacobs “The Many-Sided Frederick Jackson Turner” The Western Historical Quarterly (October 1970)
Putnam “The Turner Thesis and the Westward Movement: A Reappraisal” The Western Historical Quarterly (October 1976)
I personally am particularly fond of Howard Lamar's presidential address to the Western History Association in 1973, which he used to politely call his peers to the carpet for clinging to the frontier thesis for 75 years after it was clearly outdated. (See Lamar, "Persistent Frontier: The West in the Twentieth Century" Western Historical Quarterly, Vol. 4, No. 1 (Jan., 1973), pp. 4-25.). It was well past time, Lamar argued, for historians of the West to consider race, class, gender, environment, government, and myriad other analytical approaches rather than continuing to argue over Turner's outdated, narrow, and arguably blindingly inaccurate thesis.
All that said, there are of course many competing interpretations of the impact of the closing of the frontier. Roderick Nash's classic Wilderness in the American Mind (Yale UP, 1967) posits the cultural impact was largely as a source of anxiety. If, as Turner argued, the defining experience of America was no longer available, what next? For Nash, the links to the imperial enterprise (e.g. the Spanish-American War, annexing the Philippines, etc.) were evidence of an attempt to establish a new colonial frontier. The celebration and preservation of "frontier experiences," as reflected in the rise of national park/forest conservation, the Wild West shows, rodeo, etc., are evidence not only of nostalgia but of an attempt to preserve what vestiges remained of the frontier. And of course Teddy Roosevelt's hyper-masculine language and actions-- including his own "frontier period" spent on a ranch in the Dakotas, captured in his book Ranch Life and the Hunting Trail (1883) as well as his general "strenuous life" aesthetic --reflected this both before and after Turner's essay was published. Nash even understood the rise of the Boy Scout movement in the US, with its fetishization of Native American symbols and fake rituals, as part of this cultural response to the closing of the frontier.
If you read any of the post-war neo-Turnarians, or even someone like Michael Steiner, who in the 1990s sought to salvage Turner from the wastebin of the New Western History (NWH) (see "From Frontier to Region: Frederick Jackson Turner and the New Western History" Pacific Historical Review, Vol. 64, No. 4 (Nov., 1995), pp. 479-501), you'll find many people who felt he had not only worthwhile things to say, but that his frontier/sectional approach was a legitimate precursor to the place-based social history the young turks of the NWH movement were calling for in the late 1980s-early 1990s as the entire field went through an upheaval not seen since Turner himself was alive. Ultimately the "consequences" of the closing of the frontier are still under debate, since many don't accept the premise of the frontier in the first place, and others argue endlessly about its impacts. As David Wroebel wrote in 1996, as the New Western History was coming of age, "Frederick Jackson Turner's frontier thesis has proven hard to bury, and his ghost even more difficult to exorcise." (See "Beyond the Frontier-Region Dichotomy" Pacific Historical Review, Vol. 65, No. 3 (Aug., 1996), pp. 401-429.)
Of course no switch was flipped in 1890; there were still public lands open to homesteading and mining claims well into the 20th century. And while successive waves of white settlement transformed the trans-Mississippi West in the 19th century one can easily argue that the federal presence in the 20th c. West was an even greater source of transformation, as were the multiple waves of urbanization that continue today (ironically making the West the most urban region of the country). So before we talk about consequences of the "closing of the frontier" we have to agree there was a frontier, that it did indeed close, and that we can pinpoint that event closely enough in time to link it to some result. While Turner did all three of those in a single essay, historians have been arguing about his premise and conclusions ever since. At this juncture I think I'd personally argue that Turner's essay itself yielded far greater long-term consequences than the processes he claimed to have uncovered.
Frederick Jackson Turned borrowed heavily from the Italian economist Achille Loria (himself borrowing from Egisto Rossi). The ideas underpinning this complex international stew of ideas focused on the economics of free land - that was the escape value they all were discussing: the vast amounts of free land allowed for young, disenfranchised members of Eastern society to take the risk, go West, and create a new prosperous life. It had less to do with frontiers filled with opponents to fight than it did with the simple economics of opportunity to form a new life. These three scholars - with Turner becoming the leading American/English language proponent - looked at this unique situation as central to the success of the American democracy. As long as free land was available, the USA would be less inclined to form rigid classes with little chance for advancement. The American dream could survive because of this escape valve.
Turner's lecture pointed to the US Census's declaration that the frontier was closed by 1890. Turner suggested that this moment would mark an important transition in the American experience because the escape value would no longer exist.
Some of this concept, I am certain, could be applied to other times and places where frontiers of sorts were available for expansion.
I hope this helps; if you need more, let me know.