Diogenes and Alexander the Great were philosophical cousins. They each inherited some of their ideas from Socrates, their common philosophical ancestor.
Diogenes studied under Antisthenes who studied under Socrates, which makes Diogenes a kind of philosophical grandchild of Socrates. Alexander studied under Aristotle who studied under Plato who studied under Socrates, which makes Alexander a philosophical great grandchild of Socrates.
The difference in their approaches to the parochialism of the polis is due in part to their different resources rather than any serious differences in their philosophical ancestry.
Antisthenes, whose approach to philosophy may have been much closer to that of Socrates than Plato's was, was comparatively down to earth and skeptical. Diogenes himself had few resources - he was just one man, and he gave away almost everything he had - so it only makes sense that he would opt for the minimal approach to transcending his parochialism by changing himself.
Plato, though, was always more poet than philosopher, and his approach to philosophy was considerably less grounded than that of Antisthenes, more abstract and metaphorical. Plato was horrified when Athens condemned Socrates to death, and he retreated still further into his ideal and imaginary worlds. He transferred his loyalties from his polis of Athens to an imaginary philosophical republic of his own devising. Remarkable as Plato was, his ideal Republic may have been the first rigorous description of a totalitarian state. The worst excesses of the Inquisition and the Holocaust owe something to Plato's republic, as those totalitarian movements struggled to purify their own republics according to abstract theories about who should and should not exist in reality, much like Plato had done in theory.
Aristotle, Plato's most preeminant student, learned not just from Plato's strengths but also from his weaknesses. Instead of prescribing, he studied. Like Diogenes he was one of Plato's most severe critics. When Aristotle turned his attentions to the polis, instead of poetically theorizing about perfection, he studied the varieties of governments the polis has had over time and explored the reasons for their successes and failures. He systematically and empirically analyzed the polis to figure out what makes it work and how to make it work best. As far as we know he wrote little or nothing about Socrates's idea of the citizen of the world, perhaps because he had insufficient examples to study - Aristotle preferred the study of reality to speculation about possibilities - but it's pretty clear he talked about the idea because of what his student did with it.
Like Diogenes, Alexander inherited Socrates's ideas about the cosmopolis - the cosmos as the proper focus of our loyalty - but because of his superior resources he didn't stop by changing his own attitude about the Greek polis.
He decided to change everyone else's too. He decided to create a cosmopolitan world.