Despite his journey from Krypton to Earth granting Superman one of the world's most iconic immigrant narratives, his origin was at one point altered to rob him of this status for a decade. This happened in 1986's Man of Steel #1, and saw Superman explicitly reject his Kryptonian heritage.
The retelling of Superman's origin story starts off familiar like so many retellings before and after it. Krypton is a doomed planet, with no time left to save it or evacuate its people. But Jor-El, Superman's father, sees a way to save his son. In perhaps a minor change to the Superman mythos, Jor-El purposely locks Kal-El’s rocket onto America, specifically Kansas. Superman's mother is appalled at how the planet's men apparently work on untouched soil with their shirts off, exposing their hairy chests to the air. But Jor-El sees Earth's uncultivated freedom as a fresh start for his son.
In this version of Krypton, it appears that Kryptonians have a system to grow children in an artificial matrix placed inside rooms called gestation chambers. By law, Jor-El is allowed to remove the matrix because he is the baby's father, but this is something that has not been done for centuries. Jor-El sees no choice, though, as the planet is being destroyed. However, this technological change means that when Jor-El sends his son to Earth, Kal-El has not technically been born yet. When he lands, and emerges from the matrix, he is 'born' on Earth and becomes an American citizen, as his adopted father points out. This minor detail removes Superman's status as an immigrant to Earth, making it his native planet instead. A few issues later, in issue 6 (written and drawn by John Byrne, as was the rest of the series), Superman states, "I may have been conceived out there in the endless depths of space... but I was born when the rocket opened on Earth, in America... Krypton bred me, but it was Earth that gave me all I am. All that matters."
It's interesting to see how a story like this contrasts with other Elseworld-type stories that play around with Superman's immigration status. A notable tale is Mark Millar's Red Son. That story focuses on Superman being an immigrant and adopting the values not of America but the Soviet Union. Another more recent example is Tom Taylor's Dark Knights of Steel. In that series, Superman is still an immigrant, but he actually arrives on his adopted planet with his birth parents. So although he's an immigrant, his parents were still able to instill in him the values of Krypton.
It would take ten years for DC's official continuity to once again state that Superman was born on Krypton - a fact which remains relevant to the character to this day. While Man of Steel takes great lengths to frame Superman as an explicitly American hero, more recent stories have placed greater emphasis on his identity as the last son of Krypton, and on the complex relationship he has with his original and adopted planets. This has only deepened with the arrival of Jon Kent - Clark's son, who is native to Earth but possesses Kryptonian powers.
Ultimately, Superman's status as an immigrant to Earth has become a huge facet of his place in pop culture and the ongoing comics. While Man of Steel's Krypton is a fascinating sci-fi location, it's important that Superman feels a true connection to it, even as Jonathan and Martha Kent made sure that Earth is just as much his home.