Much of the lingo of modern protest progressivism, I think, derives from the Civil Rights era of the '60s, and the irony of the phrase "right side of history" is that it makes most sense when deployed in the context of a moral worldview that believes in eternal verities and absolutes to which mortal society can get closer, or fall back from. When Martin Luther King said, "The arc of the moral universe is long, but it bends towards justice," he was speaking out of a strong Christian sensibility (albeit one that he himself fell short of in other ways, but then we all do that) that justice was a real thing that human societies which genuinely sought the good could move closer to.
Ironically, however, that quote itself is a rhetorical paraphrase of part of an abolitionist sermon by minister Theodore Parker in 1853, and what Parker actually said was, "I do not pretend to understand the moral universe. The arc is a long one. My eye reaches but little ways. I cannot calculate the curve and complete the figure by experience of sight. I can divine it by conscience. And from what I see I am sure it bends toward justice." Most tellingly, Parker's original words lack the implication of inevitability that King's rephrase suggests; King's inadvertent genius was to mix the Christian surety of righteousness with the Marxist conviction of historical inevitability, which gave too many post-60s progressivists a fatal certainty that nothing done in the name of social justice could really be all that counterproductive as long as it kept the Vision alive.
That parallels the March of Progress, a famous piece of art by Rudolph Zallinger showing apes becoming proto-humans, who in turn become modern humans. The idea was immensely imitable, with endless variations popping up; for instance, some showed humans starting as fish and then transforming through the other Linnean classes. The original and its imitators appear to be pro-evolution, because yes we did evolve from apes, and yes we're all cladistically fish, but the way it's presented is really an attempt to place evolution within a religious context. The ideas that there are lower and higher animals, that living creatures form a progression, and particularly that modern humanity is the end result of creation, which the left-right progression suggests, are rooted in medieval Christian concepts like the Great Chain of Being as much as they are in
The Origin of Species. It's deceptive and anti-scientific, because evolution doesn't progress toward anything. More complexity does evolve, but older forms still thrive; we have tons of bacteria, insects, and crocodiles on the planet, after all, and they continue evolving. Humans are just another animal; we are neither the goal nor the apex of evolution, just a new leaf. We do have a unique combination of features, but that's because of a quirky mix of circumstances, not destiny. Teleology is wrong, and the line or arrow is simply a misleading metaphor; a radiation is better.
The idea of the "right side of history" is a similar throwback to the musty thinking of the middle ages. But I'd argue its immediate inspirations go back further than the Civil Rights era, because the idea of stages of history, inevitably progressing from one to the next, is a concept that comes from the early socialists. While Marx tried to distance himself from the early socialists by dismissing them as "utopian" and pretending to be scientific, he did adopt many of their ideas, including the stages of history. Since Marx still looms large in leftist thinking today, I'd guess his writings are at least as likely a direct source as MLK.
This would be another key distinction between progressivism and conservatism as I understand them: in conservatism the ends alone can never justify the means, and it is possible to disagree upon means even while one agrees upon ends. In modern progressivism, by contrast, any objection to means is always (or at least in the majority of my observation and experience) taken to represent a rejection of the ends; if you don't value a goal enough to consider "by any means necessary" a valid approach to realizing it, you don't really value that goal, goes the thinking.
There is a certain insistence in progressivism, that's absent in conservatism. Everything is an emergency and must be corrected
now vs. things have their time. Anyone who opposes us in any way is the enemy vs. let it be. "Everything is political", and the intrusion of politics into every form of escapism vs. not talking about politics at the dinner table. We must have the power to fix everything vs. the state has limits, and is not always to be trusted. More generally, outrage vs. politeness.