Lindsay never switched parties like Bloomberg. He was what we used to call a Rockefeller Republican. Your comment is off base.
But even as these problems were ballooning, Mr. Lindsay, urged on by a coterie of ambitious political aides, was becoming a Democrat and running for president. His 1972 race for the Democratic presidential nomination barely got off the ground. Defeated in the Florida and Wisconsin primaries, he withdrew and cast his support to Senator George S. McGovern, who was overwhelmed by Richard M. Nixon in the general election. By 1973, his last year in office, Mr. Lindsay had become a more seasoned, pragmatic mayor. Having moved from Republican congressman to Liberal mayor to Democratic presidential contender, ...,
But yes, he didn't change his party affiliation until his second term, when he knew he was termed out, re ever running again to be mayor.