Coincidence? I don't think there's a single reason: Huntsman was a diplomat before becoming a politician; Baucas had already decided to quit the Senate before taking the job; and Branstad was nearing retirement. Being ambassador arguably confers less power, but there is a sense in which it's an easier job and you don't need to win an election.
The Democratic Senator Baucas had reportedly indicated in April 2013 that he wouldn't stand for re-election in Montana, months before he was nominated as ambassador. (See Wikipedia) According to CNN his decision to leave the Senate was partly for personal reasons including his age, and partly because he was "not that happy" in the Senate - he was outside the Democrat mainstream on many issues, opposing "Obamacare" and gun control (see the Wikipedia link above for a summary of his positions). He may also have feared for his Senate seat with Montana becoming more Republican.
Terry Branstad (b. 1946) was 70 when he accepted the ambassadorship in December 2016, so age may have played a role in his desire to quit politics and pursue something that might be less stressful and more ceremonial and let him move into retirement; presumably job security wasn't uppermost in his mind. Although I can't find much to directly explain his decision.
Huntsman was still relatively young when he became ambassador in 2009, and was close to John McCain, the Republican candidate in 2008. Maybe as a Republican he felt that America was becoming more pro-Democrat following Obama's victory; maybe he was disheartened with domestic politics following McCain's loss. But he had previously worked as a diplomat, as ambassador to Singapore and as Deputy US Trade Representative, so it's not surprising he was interested in the job. He returned from China to pursue the US Presidency, but with little sign that he could become President, he's now Ambassador to Moscow and talked about for more diplomatic jobs. (Deseret News, October 10, 2018; Foreign Policy, 14 March 2018).
