Does this picture represent people, mostly Democrats, moving south?
I don't think there is any one factor. The 2012 election was a low turnout event for the reelection of a somewhat favorable African-American no-drama Democratic president versus a blandish Republican opponent that occurred when the country had in the midst of recovering from a massive economical downturn. The 2020 election was a high turnout event for the reelection of a markedly unfavorable white all-drama Republican president versus a blandish Democratic opponent that occurred when the country was in the midst of a pandemic. All of those factors, and more, contributed to the shift in that diagram.
Turnout was a key factor. Georgia turned markedly blue not because of people moving to Georgia but rather because Stacey Abrams drove a massive get-out-the-vote campaign in Georgia.
In rural midwestern states Trump energized the turnout more than did Biden. It was not people moving out. Wisconsin, Michigan, and Pennsylvania all saw higher votes for Trump in 2020 than they saw for Romney in 2012, but they also say higher votes for Biden in 2020 than they saw for Obama in 2012.
New York similarly saw higher votes for Trump in 2020 than they saw for Romney in 2012, but they also say higher votes for Biden in 2020 than they saw for Obama in 2012. It was not people moving out. Upstate New York, which is fairly rural, saw a huge increase in turnout, and that turnout favored Trump. But Trump still lost by a landslide in New York in 2020.
Utah turned bluer because there was no way to go but down from the 48% routing taken by Obama in 2012 against the first Mormon candidate for president. Trump took Utah by "only" 20.5% in 2020. That's still a landslide.